Rotor pair for a compression block of a screw machine

ABSTRACT

The invention relates to a rotor pair for a compressor block of a screw machine, wherein the rotor pair comprises a secondary rotor that rotates about a first axis and a main rotor that rotates about a second axis, wherein the number of teeth of the main rotor is 3 and the number of teeth of the secondary rotor is 4. The relative profile depth of the secondary rotor is at least 0.5, preferably at least 0.515, and at most 0.65, preferably at most 0.595. rk1 is an addendum circle radius drawn around the outer circumference of the secondary rotor and rf1 is a dedendum circle radius starting at the profile base of the secondary rotor, wherein the ratio of the axis distance of the first axis from the second axis and the addendum circle radius rk1 is at least 1.636, and at most 1.8, preferably at most 1.733.

RELATED APPLICATIONS

The present application is a divisional of U.S. patent application Ser.No. 16/530,002 filed Aug. 2, 2019, which application is divisional ofU.S. patent application Ser. No. 15/306,592, filed Oct. 25, 2016, whichapplication is a 35 U.S.C. § 371 national phase application of PCTInternational Application No. PCT/EP2015/059070, filed Apr. 27, 2015,which claims priority from German Patent Application No. 10 2014 105882.8, filed Apr. 25, 2014; the disclosures of which are herebyincorporated herein by reference in their entirety. PCT InternationalApplication No. PCT/EP2015/059070 is published in German as PCTPublication No. WO 2015/162296.

FIELD OF THE INVENTION

The invention relates to a rotor pair for a compressor block of a screwmachine, where the rotor pair consists of a main rotor that rotatesabout a first axis and a secondary rotor that rotates about a secondaxis. The invention further relates to a compressor block having acorresponding rotor pair.

BACKGROUND

Screw machines, whether this be in the form of screw compressors or inthe form of screw expanders, have been in practical use for severaldecades. Configured as screw compressors, they have supersededreciprocating piston compressors as compressors in many areas. With theprinciple of the intermeshing pair of screws, not only gases can becompressed by applying a certain amount of work. The application as avacuum pump also opens up the use of screw machines to achieve a vacuum.Finally an amount of work can also be produced by passing throughpressurized gases the other way round so that mechanical energy can alsobe obtained from pressurized gases by means of the principle of thescrew machine.

Screw machines generally have two shafts arranged parallel to oneanother on which a main rotor on the one hand and a secondary rotor onthe other hand are located. Main rotor and secondary rotor intermeshwith a corresponding screw-shaped toothed structure. Between the toothedstructures and a compressor housing which accommodates the main andsecondary rotor, a compression chamber (working chambers) is formed bythe tooth gap volumes. Starting from a suction region as the rotation ofmain and secondary rotor progresses, the working chamber is initiallyclosed and then continuously reduced in volume so that a compression ofthe medium occurs. Finally as rotation progresses, the working chamberis opened towards a pressure window and the medium is expelled into thepressure window. Screw machines configured as screw compressors differby this process of internal compression from Roots blowers which operatewithout internal compression.

Depending on the required pressure ratio (ratio of output pressure toinput pressure), various tooth number ratios are appropriate forefficient compression.

Typical pressure ratios can be between 1.1 and 20 depending on the toothnumber ratio, where the pressure ratio is the ratio of compression endpressure to suction pressure. The compression can take place in asingle- or multistage manner. Attainable final pressures can, forexample, lie in the range of 1.1 bar to 20 bar. Insofar as at this pointor hereinafter in the present application reference is made to pressureinformation in “bar”, in each case this pressure information relates toabsolute pressures.

In addition to the already mentioned function as a vacuum pump or as ascrew expander, screw machines can be used in various areas oftechnology as compressors. A particularly preferred area of applicationis the compression of gases such as, for example, air or inert gases(helium, nitrogen, . . . ). However, it is also possible, although thisimposes especially structurally different requirements, to use a screwmachine to compress refrigerants, for example for air-conditioningsystems or refrigeration applications. For the compression of gasesspecifically with higher pressure ratios, usually a fluid-injectedcompression, in particular an oil-injected compression is used; howeverit is also possible to operate a screw machine according to theprinciple of dry compression. In the lower-pressure area, screwcompressors are occasionally also designated as screw blowers.

Over the past few decades, considerable success has been achieved inregard to the manufacturability, reliability, smooth running andefficiency of screw machines. Improvements or optimizations in thiscontext frequently relate to optimizations of the efficiency dependingon number of teeth, wrap-around angle and length/diameter ratio of therotors. The incorporation of the transverse sections in the optimizationprocess has only taken place recently.

Experiments have shown that the transverse section of the rotors, inparticular the transverse section of the secondary rotor has asubstantial influence on the energy efficiency. In order to obey thetoothed structure laws, the transverse section of the secondary rotormust find its correspondence in the transverse section of the mainrotor. The profile of the rotor in a plane perpendicular to the axis ofthe rotor is here designated as transverse section. Various types oftransverse section generation such as, for example, rotor- or rack-basedtransverse section generating methods are now known from the prior art.If a specific process has been decided upon, a first draft transversesection is generated in a first step. This is conventionally furtheroptimized in a plurality of successive (revising) steps according tovarious criteria.

Here both the optimization aims per se (energy efficiency, smoothrunning, low costs) and also the fact that the improvements of oneparameter in some cases necessarily result in a deterioration of anotherparameter, are known. However, there is a lack of a specific solution asto how a good overall optimization result (i.e. a compromise between thevarious individual parameter optimizations) can be achieved.

Some optimization approaches which are known in the prior art with aview to improving the energy efficiency, smooth running and costs willbe explained as an example hereinafter. Furthermore, problems which canarise here will also be mentioned.

1 Energy Efficiency

The energy efficiency of compressor blocks can advantageously beinfluenced in a known manner by minimizing the internal leakages in thecompressor block and in particular by reducing the gap between mainrotor and secondary rotor. Specifically here a distinction should bemade between the profile gap and the blow hole:

-   -   Via the profile gap the pressure-side working chambers have        direct communication to the suction side and therefore the        greatest possible pressure difference for backflows.    -   Consecutive working chambers are interconnected via a        theoretically unnecessary passage which is designated as blow        hole. In some cases this is also designated as head rounding        opening. This blow hole is obtained through the head rounding of        the profiles, in particular the profile of the secondary rotor.    -   Pressure-side working chambers are connected to the respectively        adjacent working chamber via pressure-side blow holes,        suction-side working chambers are connected to the respectively        adjacent working chambers via suction-side blow holes. Unless        specified otherwise, the term “blow hole” is to be understood        hereinafter as “pressure-side blow hole”.

Ideally, in order to minimize internal leakages, a short profile gaplength should be combined with a small (pressure-side) blow hole.However, the two quantities behave fundamentally contrarily. That is,the smaller the blow hole is modelled, the larger the profile gap lengthmust be. Conversely, the blow hole becomes larger, the shorter is theprofile gap length. This is explained, for example, by Helpertz in hisdissertation “Method for the stochastic optimization of screw rotorprofiles”, Dortmund, 2003, on page 162.

The requirement for a short profile gap length can be achieved in aknown manner with a flat profile with a relatively small relativeprofile depth of the secondary rotor. Whether a profile is designed tobe rather flat (small profile depth) or deep (large profile depth) canbe clearly quantified here by means of the so-called “relative profiledepth of the secondary rotor” which relates the difference betweenaddendum and dedendum circle radius to the addendum circle radius of thesecondary rotor. The higher is the value, the more compact is thecompressor block and for example, has more quantity delivered than acomparable compressor block with the same external dimensions.

Profiles designed to be very flat accordingly have a poor utilization ofinstallation volume, i.e. they result in large compressor blocks withcomparatively high material expenditure or comparatively highmanufacturing costs.

Pressure-side blow holes as described above must not be designed to betoo large in order to minimize the return flow of already compressedmedium in preceding working chambers (i.e., in lower-pressure workingchambers). Such return flows increase the energy expenditure for theoverall conveying capacity achieved and result in an undesirableincrease in the temperature and pressure level during compression whichoverall reduces the efficiency. The area of the blow hole (blow holearea) can be kept small whereby the head roundings of the profiles inthe transverse section are designed to be small. Specifically, this canbe achieved by a strong curvature in the head region of the leadingtooth flank of the secondary rotor and in the head region of thetrailing tooth flank of the main rotor. However, the stronger is thiscurvature, the more rapidly production-technology limiting regions arereached since this for example results in high wear on profile millersand profile grinding disks during the manufacture of main rotor andsecondary rotor.

Suction-side blow holes on the other hand do not have a negativeinfluence on the energy efficiency since only working chambers in thesuction region are interconnected via these at the same pressure.

Another cause of efficiency-reducing internal leakages is the so-calledchamber interstitial volume which can form during expulsion of the lastworking chamber (i.e. the working chamber in which the highest pressureprevails) into the pressure window. The working chamber then no longerhas a connection to the pressure window from a certain rotational angleposition of the rotors. A so-called chamber interstitial volume remainsbetween the two rotors and the pressure-side housing end wall.

This chamber interstitial volume is disadvantageous because the enclosedcompressed medium can no longer be expelled into the pressure window andis even further compressed during the further rotation of the rotors,which leads to an unnecessarily high power consumption (for theover-compression), an unnecessarily high additional heat input,evolution of noise and a reduction in the lifetime, in particular of theroller bearings of the rotors. In addition, a deterioration in thespecific power is caused by the fact that the fraction enclosed in thechamber interstitial volume is returned to the suction side after theover-compression and therefore is no longer available to the compressedair user. In the case of oil-injected compressors, incompressible oil isadditionally in the chamber interstices and is squeezed.

2 Smooth Running

However, other properties such as, for example, the smooth running alsohave a decisive influence on a good profile for main rotor or secondaryrotor.

In addition to good osculation of the flanks and low relative speedsbetween the tooth flanks of main and secondary rotor, the division ofthe drive torque between the two rotors also has a decisive influence onthe two rotors. An unfavourable distribution is known to frequentlyresult in so-called rotor rattling of the secondary rotor in which thesecondary rotor has undefined flank contact with the main rotor and thesecondary rotor consequently alternately has contact with the leadingand the trailing main rotor flank. If the two rotors are held at adistance by means of a synchronous transmission, the aforesaid rotorrattling is necessarily displaced into the synchronous transmission.Good smooth running not only ensures low sound emissions from thecompressor block but for example also provides for a lessvibration-prone compressor block, a long lifetime of the roller bearingsand low wear in the tooth structure of the rotors.

3 Costs

In particular, the manufacturability and the degree of utilization ofthe installation volume have an effect on the material and manufacturingcosts of screw compressor blocks. Compact compressor blocks with a highutilization of installation volume are achieved by a large tooth gapvolume which in turn depends on the profile depth and the tooththickness.

The further the relative profile depth is increased, the higherutilization of installation volume is achieved but at the same time, therisk of problems with running properties and manufacturability ishigher:

-   -   With increasing profile depth, in particular the tooth profiles        of the secondary rotor will necessarily become increasingly        thinner and consequently increasingly flexible. This makes the        rotors increasingly temperature-sensitive and when viewed        overall, has an unfavourable effect on the gaps in the        compressor block. The gaps have an appreciable influence on the        internal leakages, i.e. return flows from higher-pressure        compression chambers in the direction of the suction side, and        can thus cause a deterioration in the energy efficiency of the        compressor block.    -   Furthermore, in the case of flexible teeth the difficulties with        rotor manufacture increase.        -   Thus for example, there is an increased risk that the            requirements in particular for the shape tolerances, which            are already high in any case, cannot be adhered to.        -   Furthermore, flexible teeth require lower feed and            intersection speeds both during profile milling and also            during subsequent profile grinding and thus increase the            processing time and consequently the manufacturing costs.    -   An increasing profile depth also has the result that the rotor        per se becomes more flexible. The more flexible the rotors are        designed, the more the risk increases that the rotors start        running amongst one another or in the compressor housing. In        order to ensure operating safety even at high temperatures or at        high pressures, the gaps must consequently have larger        dimensions. This in turn has a negative influence on the energy        efficiency of the compressor block.

SUMMARY

The above explanations are intended to show that an optimization of theindividual characteristics each for itself is less expedient but for agood overall result a compromise must be found between the various (andpartly contradictory) requirements.

The theoretical calculation principles for producing screw rotorprofiles have already been discussed on many occasions in the literatureand also describe general criteria for good transverse section profiles.For example, rotor profiles can be created and modified using thecomputer program developed by Grafinger (post-doctoral thesis“Computer-assisted development of flank profiles for special toothstructures of screw compressors”, Vienna, 2010).

In his thesis “Method for the stochastic optimization of screw rotorprofiles”, Dortmund 2003, Helpertz is concerned with the automatedoptimization starting from a draft with regard to differently weightedcharacteristics.

Accordingly it is the object of the present invention to provide a rotorpair for a compressor block of a screw machine which is characterized byhighly smooth running and a particular energy efficiency with highoperating safety and acceptable production costs.

This object is solved with a rotor pair. Advantageous embodiments arespecified in the subclaims. Further, the object is also solved with acompressor block comprising a suitably configured rotor pair.

The rotor geometry is substantially characterized by the shape of thetransverse section as well as by the rotor length and the wrap-aroundangle, cf. “Method for the stochastic optimization of screw rotorprofiles”, Thesis by Markus Helpertz, Dortmund 2003, pp. 11/12.

In a transverse sectional view, secondary rotor or main rotor have apre-determined, frequently different number of identically configuredteeth per rotor. The outermost circle drawn through the axis C1 or C2via the apex points of the teeth is designated as addendum circle ineach case. A dedendum circle is defined by the points of the outersurface of the rotors nearest to the axis in transverse section. Theribs are designated as teeth of the rotor. The grooves (or recesses) areaccordingly designated as tooth gaps. The surface of the tooth at andover the dedendum circle defines the tooth profile. The contour of theribs defines the course of the tooth profile. Foot points F1 and F2 andan apex point F5 are defined for the tooth profile. The apex point F5 orH5 is defined by the radially outermost point of the tooth profile. Ifthe tooth profile has a plurality of points with the same maximum radialdistance from the central point defined by the axis C1 or C2, the toothprofile therefore follows at its radially outermost end a circular arcon the addendum circle, the apex point F5 lies precisely at the centreof this circular arc. A tooth gap is defined between two adjacent apexpoints F5.

The points radially nearest to the axis C1 or C2 between an observed andthe respectively adjacent tooth define foot points F1 and F2. Here italso holds for the case that a plurality of points come equally close tothe axis C1 or C2, i.e. the tooth profile at its lowest point followsthe dedendum circle in sections, that the corresponding foot point F1 orF2 then lies on the half of this circular arc lying on the dedendumcircle.

Finally, as a result of the intermeshing of main rotor and secondaryrotor, a pitch circle is defined both for the secondary rotor and alsofor the main rotor. In screw machines and also in gear wheels orfriction wheels, there are always two circles in the transverse sectionof the toothed structure which roll against one another during themovement. These circles on which in the present case main rotor andsecondary rotor roll against one another are designated as respectivepitch circles. The pitch circle diameter of main rotor and secondaryrotor can be determined with the aid of axial distance and tooth numberratio.

On the pitch circles the circumferential speeds of main rotor andsecondary rotor are identical.

Finally tooth gap areas between the teeth and the respective addendumcircle KK are defined, namely tooth gap area A6 between the profilecourse of the secondary rotor NR between two adjacent apex points F5 andthe addendum circle KK₁ or an area A7 as tooth gap area between theprofile course of the main rotor (HR) between two adjacent apex pointsH5 and the addendum circle KK₂.

The tooth profile of the secondary rotor (but also of the main rotor)has a leading tooth flank in the direction of rotation and a trailingtooth flank in the direction of rotation. In the secondary rotor (NR)the leading tooth flank is hereinafter designated by F_(V) and thetrailing tooth flank by F_(N).

The trailing tooth flank F_(N) in its section between addendum circleand dedendum circle forms a point at which the curvature of the courseof the tooth profile changes. This point is hereinafter designated as F8and divides the trailing tooth flank F_(N) into a convexly curvedfraction between F8 and the addendum circle and a concavely curvedfraction between the dedendum circle and F8. Small-part profilevariations, possibly due to sealing strips or due to other local profilerestructurings are not taken into account when considering thepreviously described change of curvature.

In addition to the pure transverse section, for the three-dimensionalconfiguration, the following terms or parameters are definitive for arotor, in particular the secondary rotor: firstly the wrap-around angleϕ is defined. This wrap-around angle is the angle through which thetransverse section is turned from the suction-side to the pressure-siderotor end face, cf. on this matter also the more detailed explanationsin connection with FIG. 8.

The main rotor has a rotor length L_(HR) which is defined as thedistance of a suction-side main-rotor rotor end face to a pressure-sidemain-rotor rotor end face. The distance of the first axis C1 of thesecondary rotor to the second axis C2 of the main rotor running parallelto one another is hereinafter designated as axial distance a. It ispointed out that in most cases the length of the main rotor L_(HR)corresponds to the length of the secondary rotor L_(NR), where in thecase of the secondary rotor the length is also understood as thedistance of a suction-side secondary-rotor rotor end face to apressure-side secondary-rotor rotor end face. Finally a rotor lengthratio L_(HR)/a is defined, i.e. a ratio of the rotor length of the mainrotor to the axial distance. The ratio L_(HR)/a is in this respect ameasure for the axial dimensioning of the rotor profile.

The line of engagement or the profile gap is formed by the cooperationof main rotor and secondary rotor with one another. In this case, theline of engagement is obtained as follows: the tooth flanks or mainrotor and secondary rotor contact one another in a backlash-free toothedstructure depending on the rotational angle position of the rotors atcertain points. These points are designated as engagement points. Thegeometric location of all the engagement points is the line ofengagement and can already be calculated in two dimensions by means ofthe transverse section of the rotors, cf. FIG. 7 j.

In the transverse sectional view, the line of engagement is divided bythe connecting line between the two central points C1 and C2 into twosections and specifically into a (comparatively short) suction-side anda (comparatively long) pressure-side section.

If the wrap-around angle and the rotor length (=distance between thesuction-side end face and the pressure-side end face) are additionallyspecified, the line of engagement can also be expandedthree-dimensionally and corresponds to the line of contact of main rotorand secondary rotor. The axial projection of the three-dimension line ofengagement on the transverse sectional plane in turn gives thetwo-dimensional line of engagement illustrated by means of FIG. 7j . Theterm “line of engagement” is used in the literature both for thetwo-dimensional and the three-dimensional analysis. Hereinafter, unlessspecified otherwise, “line of engagement” is understood however as thetwo-dimensional line of engagement, i.e. the projection onto thetransverse section.

The profile engagement gap is defined as follows: in a real compressorblock of a screw machine, there is a gap between the two rotors with theinstalled axial spacing of main rotor and secondary rotor. The gapbetween main rotor and secondary rotor is designated as profileengagement gap and is the geometrical location of all the points atwhich the two paired rotors contact one another or have the smallestdistance from one another.

Through the profile engagement gap the compressing and the expellingworking chambers are in communication with chambers which still havecontact with the suction side. Therefore the total maximum pressureratio is present at the profile engagement gap. Through the profileengagement gap, already compressed working fluid is transported back tothe suction side and thus reduces the efficiency of the compression.Since the profile engagement gap in a backlash-free toothed structurewould comprise the line of engagement, the profile engagement gap isalso designated as “quasi-engagement line”.

Blow holes between working chambers are formed by head roundings of theteeth of the profile. Via blow holes the working chambers are connectedto the preceding and following working chambers so that (in contrast tothe profile engagement gap) only the pressure difference from oneworking chamber to the next working chamber is present at the blow hole.

Furthermore, as is known, certain rotor pairs are usual in screwmachines, for example a rotor pair in which the main rotor has threeteeth and the secondary rotor has four teeth or a rotor pair in whichthe main rotor has four teeth and the secondary rotor has five teeth orfurthermore a rotor pair geometry in which the main rotor has five teethand the secondary rotor has six teeth. For different areas ofapplication or intended uses, rotor pairs or screw machines havingdifferent tooth number ratios are possibly used. For example, rotor pairarrangements having a tooth number ratio of 4/5 (main rotor with fourteeth, secondary rotor with five teeth) are used as a suitable pair foroil-injected compression applications in moderate pressure ranges.

In this respect, the tooth number or the tooth number ratio predefinesdifferent types of rotor pairs and resulting from this, different typesof screw machines, in particular screw compressors.

For a screw machine or a rotor pair with three teeth in the main rotorand four teeth in the secondary rotor, a geometry having the followingspecifications is claimed, which can be deemed to be particularlyenergy-efficient:

A relative profile depth of the secondary rotor is configured with

${PT_{rel}} = \frac{{rk_{1}} - {rf_{1}}}{rk_{1}}$

where PT_(rel) is at least 0.5, preferably at least 0.515, and at most0.65, preferably at most 0.595, wherein rk₁ is an addendum circle radiusdrawn around the outer circumference of the secondary rotor and rf₁ is adedendum circle radius starting at the profile base of the secondaryrotor. Furthermore, the ratio of the axis distance a of the first axisC1 from the second axis C2 and the addendum circle radius rk₁

$\frac{a}{{rk}_{1}}$

is specified so that

$\frac{a}{{rk}_{1}}$

is at least 1.636 and at most 1.8, preferably at most 1.733, whereinpreferably the main rotor is configured with a wrap-around angle ϕ_(HR)for which it holds that 240°≤ϕ_(HR)≤360°, and wherein preferably for arotor length ratio L_(HR)/a it holds that:

1.4≤L _(HR) /a≤3.4,

wherein the rotor length ratio is formed from the ratio of the rotorlength Lim of the main rotor and the axis distance a and the rotorlength L_(HR) of the main rotor is formed by the distance of asuction-side main-rotor rotor end face to an opposite pressure-sidemain-rotor rotor end face.

For a screw machine or a rotor pair with four teeth in the main rotorand five teeth in the secondary rotor, a geometry having the followingspecifications is claimed, which can be deemed to be particularlyenergy-efficient: a relative profile depth of the secondary rotor isconfigured with

${PT_{rel}} = \frac{{rk_{1}} - {rf_{1}}}{rk_{1}}$

wherein PT_(rel) is at least 0.5, preferably at least 0.515, and at most0.58, wherein rk₁ is an addendum circle radius drawn around the outercircumference of the secondary rotor and rf₁ is a dedendum circle radiusstarting at the profile base of the secondary rotor. Furthermore theratio of the axis distance a of the first axis C1 from the second axisC2 and the addendum circle radius rk₁

$\frac{a}{{rk}_{1}}$

is specified so that

$\frac{a}{{rk}_{1}}$

is at least 1.036 and at most 1.836, preferably at most 1.782, whereinpreferably the main rotor is configured with a wrap-around angle ϕ_(HR)for which it holds that 240°≤ϕ_(HR)≤360°, and wherein preferably for arotor length ratio L_(HR)/a it holds that:

1.4≤L _(HR) /a≤3.3,

wherein the rotor length ratio is formed from the ratio of the rotorlength Um of the main rotor and the axis distance a and the rotor lengthL_(HR) of the main rotor is formed by the distance of a suction-sidemain-rotor rotor end face to an opposite pressure-side main-rotor rotorend face.

For a screw machine or a rotor pair with five teeth in the main rotorand six teeth in the secondary rotor, a geometry having the followingspecifications is claimed, which can be deemed to be particularlyenergy-efficient:

A relative profile depth of the secondary rotor is configured with

${PT_{rel}} = \frac{{rk_{1}} - {rf_{1}}}{rk_{1}}$

wherein PT_(rel) is at least 0.44 and at most 0.495, preferably at most0.48, wherein rk₁ is an addendum circle radius drawn around the outercircumference of the secondary rotor and rf₁ is a dedendum circle radiusstarting at the profile base of the secondary rotor. Furthermore theratio of the axis distance a of the first axis C1 from the second axisC2 and the addendum circle radius rk₁

$\frac{a}{{rk}_{1}}$

is specified so that

$\frac{a}{{rk}_{1}}$

is at least 1.74, preferably at least 1.75 and at most 1.8, preferablyat most 1.79, wherein preferably the main rotor is configured with awrap-around angle ϕ_(HR) for which it holds that 240°≤ϕ_(HR)≤360°, andwherein preferably for a rotor length ratio L_(HR)/a it holds that:

1.4≤L _(HR) /a≤3.2,

wherein the rotor length ratio is formed from the ratio of the rotorlength L_(HR) of the main rotor and the axis distance a and the rotorlength L_(HR) of the main rotor is formed by the distance of asuction-side main-rotor rotor end face to an opposite pressure-sidemain-rotor rotor end face.

If the values for the relative profile depth on the one hand and theratio of axis distance to the addendum circle radius of the secondaryrotor on the other hand for the given teeth-number ratios lie in thespecified advantageous ranges in each case, the basic conditions for agood secondary rotor profile or a good cooperation of the secondaryrotor profile and main rotor profile are created, in particular aparticularly favourable ratio of blow hole area to profile gap length ismade possible. With regard to the definitive parameters, reference isadditionally made to the illustration in FIG. 7a for all the addressedtooth number ratios. The relative profile depth of the secondary rotoris a measure for how deeply the profiles are cut. With increasingprofile depth, the installation volume utilization increases for examplebut at the expense of the flexural rigidity of the secondary rotor. Forthe relative profile depth of the secondary rotor it holds that:

${PT}_{rel} = {\frac{{rk}_{1} - {rf}_{1}}{{rk}_{1}} = {\frac{{PT}_{1}}{{rk}_{1}} = {\frac{{rk}_{1} - \left( {a - {rk}_{2}} \right)}{{rk}_{1}} = {1 - \frac{a - {rk}_{2}}{{rk}_{1}}}}}}$

where PT₁=rk₁−rf₁ and rf₁=a−rk₂.

In this respect, there is a relationship with the ratio of

$\frac{a}{{rk}_{1}},$

axis distance a to the secondary rotor addendum circle radius rk₁.

The specified values for the rotor length ratio L_(HR)/a and thewrap-around angle ϕ_(HR) constitute advantageous or expedient values forthe respectively given tooth number ratio in order to specify anadvantageous rotor pair in the axial dimension.

1. Preferred Embodiments for a Rotor Pair with a Tooth Number Ratio of3/4

Preferred embodiments are set out hereinafter for a rotor pair with atooth number ratio 3/4, i.e. for a rotor pair in which the main rotorhas three teeth and the secondary rotor has four teeth:

A first preferred embodiment provides that in a transverse sectionalview, circular arcs B₂₅, B₅₀, B₇₅ running within a secondary rotor toothare defined, the common centre point of which is given by the axis C1,wherein the radius r₂₅ of B₂₅ has the value r₂₅=rf₁+0.25*(rk₁−rf₁), theradius r₅₀ of B₅₀ has the value r₅₀=rf₁+0.5*(rk₁−rf₁), and the radiusr₇₅ of B₇₅ has the value r₇₅=rf₁+0.75*(rk₁−rf₁), and wherein thecircular arcs B₂₅, B₅₀, B₇₅ are each delimited by the leading toothflank F_(V) and trailing tooth flank F_(N), wherein tooth thicknessratios are defined as ratios of the arc lengths b₂₅, b₅₀, b₇₅ of thecircular arcs B₂₅, B₅₀, B₇₅ with ε₁=b₅₀/b₂₅ and ε₂=b₇₅/b₂₅ and thefollowing dimension is adhered to: 0.65≤ε₁<1.0 and/or 0.50≤ε₂≤0.85,preferably 0.80≤ε₁<1.0 and/or 0.50≤ε₂≤0.79.

The aim is to combine a small blow hole with short length of the profileengagement gap. However the two parameters behave in a contrary manner,i.e. the smaller the blow hole is modelled, the larger the length of theprofile engagement gap necessarily becomes. Conversely the blow holebecomes larger, the shorter is the length of the profile engagement gap.In the claimed ranges a particularly favourable combination of the twoparameters is achieved. At the same time a sufficiently high flexuralrigidity of the secondary rotor is achieved. Furthermore, advantages areestablished as far as the chamber expulsion is concerned and for thesecondary rotor torque. With regard to the illustration of theparameters, reference is additionally made to FIG. 7 c.

A further preferred embodiment provides that in a transverse sectionalview, foot points F1 and F2 are defined between the observed tooth ofthe secondary rotor (NR) and the respectively adjacent tooth of thesecondary rotor and an apex point F5 is defined at the radiallyoutermost point of the tooth, wherein a triangle D_(z) is defined by F1,F2 and F5 and wherein in a radially outer region, the tooth projectsbeyond the triangle D_(z) with its leading tooth flank F_(V) formedbetween F5 and F2 with an area A1 and with its trailing tooth flankF_(N) formed between F1 and F5 with an area A2 and wherein 8≤A2/A1≤60 ismaintained.

The tooth sub-area A1 at the leading tooth flank FV of the secondaryrotor has a substantial influence on the blow hole area. The toothsub-area A2 at the trailing tooth flank F_(N) of the secondary rotor onthe other hand has a substantial influence on the length of the profileengagement gap, the chamber expulsion and the secondary rotor torque.For the tooth sub-area ratio A2/A1 there is an advantageous range whichenables a good compromise between length of the profile engagement gapon the one hand and the blow hole on the other hand. With regard to theillustration of the parameters, reference is additionally made to FIG. 7d.

In a further preferred embodiment the rotor pair comprises a secondaryrotor in which in a transverse sectional view, foot points F1 and F2 aredefined between the observed tooth of the secondary rotor (NR) and therespectively adjacent tooth of the secondary rotor, and an apex point F5is defined at the radially outermost point of the tooth, wherein atriangle D_(z) is defined by F1, F2 and F5 and wherein in a radiallyouter region of the tooth, the leading tooth flank F_(V) formed betweenF5 and F2 projects with an area A1 beyond the triangle D_(Z) and in aradially inner region is set back with respect to the triangle D_(z)with an area A3 and wherein 7.0≤A3/A1≤35 is maintained. With regard tothe illustration of the parameters, reference is additionally made toFIG. 7 d.

Furthermore, with regard to the configuration of the secondary rotor, itis considered to be advantageous if in a transverse sectional view, footpoints F1 and F2 are defined between the observed tooth of the secondaryrotor (NR) and the respectively adjacent tooth of the secondary rotor(NR) and an apex point F5 is defined at the radially outermost point ofthe tooth, wherein a triangle D_(z) is defined by F1, F2 and F5 andwherein in a radially outer region of the tooth, the leading tooth flankF_(V) formed between F5 and F2 projects with an area A1 beyond thetriangle D_(Z) and wherein the tooth itself has a cross-sectional areaA0 delimited by the circular arc B running between F1 and F2 about thecentre point defined by the axis C1 and wherein 0.5%≤A1/A0≤4.5% ismaintained. With regard to the illustration of the parameters, referenceis additionally made to FIGS. 7d and 7 e.

A further preferred embodiment provides that in a transverse sectionalview, foot points F1 and F2 are defined between the observed tooth ofthe secondary rotor (NR) and the respectively adjacent tooth of thesecondary rotor and an apex point F5 is defined is defined at theradially outermost point of the tooth, wherein the circular arc Brunning between F1 and F2 defines a tooth partition angle αcorresponding to 360°/number of teeth of the secondary rotor (NR) aboutthe centre point defined by the axis C1, wherein a point F11 is definedon the half circular arc B between F1 and F2, wherein a radial half-lineR drawn from the centre point of the secondary rotor (NR) defined by theaxis C1 through the apex point F5 intersects the circular arc B at apoint F12, wherein an offset angle β is defined by the offset of F11 toF12 viewed in the direction of rotation of the secondary rotor (NR) andwherein 14%≤δ≤25% is maintained, where

$\delta = {\frac{\beta}{\gamma}*{{100\;\lbrack\%\rbrack}.}}$

Firstly it is again clarified that the offset angle is preferably alwayspositive, i.e. the offset is always given in the direction of thedirection of rotation and not contrary to this. In this respect thetooth of the secondary rotor is curved with respect to the axis ofrotation of the secondary rotor. However, the offset should be kept in arange specified as advantageous in order to enable a favourablecompromise between the blow hole area, the shape of the engagement line,the length and the shape of the profile engagement gap, the secondaryrotor torque, the flexural rigidity of the rotors and the chamberexpulsion into the pressure window. With regard to the illustration ofthe parameters, reference is additionally made to FIG. 7 f.

It is considered to be advantageous if in a transverse sectional view,the trailing tooth flank F_(N) of a tooth of the secondary rotor (NR)formed between F1 and F5 has a convex length component of at least 45%to at most 95%.

The relatively long convex length component of the trailing tooth flankF_(N) of a tooth of the secondary rotor specified with the range allowsa good compromise between length of the profile engagement gap, chamberexpulsion, secondary rotor torque on the one hand and flexural rigidityof the secondary rotor on the other hand. With regard to theillustration of the parameters, reference is additionally made to FIG. 7g.

Preferably the secondary rotor is configured in such a manner that in atransverse sectional view, the radial half-line drawn from the axis C1of the secondary rotor (NR) through F5 divides the tooth profile into anarea component A5 assigned to the leading tooth flank F_(V) and an areacomponent A4 assigned to the trailing tooth flank F_(N) and wherein

5≤A4/A5≤14

is maintained. It should be noted once again at this point that thetooth profile is delimited radially inwards towards the C1 axis by thededendum circle FK₁. In this case, it can occur that the radialhalf-line R divides the tooth profile in such a manner that two disjointarea components with a total area component A5 which are assigned to theleading tooth flank F_(V) are formed, cf. FIG. 7g . If the apex point F5were to be offset with respect to the leading tooth flank in such amanner that the radial half-line F5 not only touches the leading toothflank F_(V) but intersects it at two points, two disjoint areacomponents assigned to the leading tooth flank with a total areacomponent A5 are again defined. The area component A4 assigned to thetrailing tooth flank F_(N) is then delimited on the one hand by theradial half-line R, in sections, namely between the two points ofintersection of the leading tooth flank F_(V) with the radial half-line,on the other hand by the leading tooth flank F_(V).

A further preferred embodiment comprises a rotor pair which ischaracterized in that the main rotor HR is formed with a wrap-aroundangle ϕ_(HR) for which it holds that: 290°≤ϕ_(HR)≤360°, preferably320°≤ϕ_(HR)≤360°.

With increasing wrap-around angle, the pressure window area can beconfigured to be larger for the same built-in volume ratio. In addition,the axial extension of the working chamber to be expelled, the so-calledprofile pocket depth, is shortened. This reduces the expulsion throttlelosses in particular at higher rotational speeds and thus enables abetter specific performance. A too-large wrap-around angle in turn has adisadvantageous effect on the installation volume and results in largerrotors.

In addition, in an advantageous embodiment a rotor pair is providedwhich is configured in such a manner and interacts with one another sothat a blow hole factor μ_(B1) is at least 0.02% and at most 0.4%,preferably at most 0.25%, wherein

$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\;\lbrack\%\rbrack}}$

arm wherein A_(B1) designates an absolute pressure-side blow hole areaand A6 and A7 designate tooth gap areas of the secondary rotor (NR) orthe main rotor (HR), wherein the area A6 in a transverse sectional viewis the area enclosed between the profile course of the secondary rotor(NR) between two adjacent apex points F5 and the addendum circle KK₁ andthe area A7 in a transverse sectional view is the area enclosed betweenthe profile course of the main rotor (HR) between two adjacent apexpoints H5 and the addendum circle KK₂.

Whereas the absolute magnitude of the pressure-side blow hole alone doesnot allow any meaningful prediction about the effect on leakage massflows, a ratio of the absolute pressure-side blow hole area AB1 to thesum of the tooth gap area A6 of the secondary rotor and the tooth gaparea A7 of the main rotor is substantially more predictive. With regardto the further illustration of the parameters, reference is additionallymade here to FIG. 7b . The lower the numerical value of μ_(B1), thesmaller is the influence of the blow hole on the operating behaviour.The pressure-side blow hole area can thus be represented independentlyof the installation size of the screw machine.

In a further preferred embodiment, a rotor pair is configured andmatched to one another in such a manner that for a blow hole/profile gaplength factor μ₁*μ_(B1) it holds that

0.1%≤μ₁*μ_(B1)≤1.72%

where

${\mu_{l} = \frac{l_{sp}}{{PT}_{1}}},$

where l_(sp) designates the length of the profile engagement gap of atooth gap of the secondary rotor and PT₁ designates the profile depth ofthe secondary rotor, where PT₁=rk₁−rf₁and

$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\;\lbrack\%\rbrack}}$

where A_(B1) designates the absolute blow hole area and A6 and A7designate the profile areas of the secondary rotor (NR) or the mainrotor (HR), wherein the area A6 in a transverse sectional viewdesignates the area enclosed between the profile course of the secondaryrotor (NR) between two adjacent apex points F5 and the addendum circleKK₁, and the area A7 in a transverse sectional view designates the areaenclosed between the profile course of the main rotor (HR) between twoadjacent apex points H5 and the addendum circle KK₂.

μ₁ designates a profile gap length factor, where a length of the profileengagement gap of a tooth gap is related to the profile depth PT₁. Thus,a measure for the length of the profile engagement gap can be specifiedindependently of the installation size of the screw machine, The lowerthe numerical value of the characteristic μ₁, the shorter is the profilegap of a tooth pitch for the same profile depth and therefore thesmaller is the leakage volume flow back to the suction side. The factorμ₁*μ_(B1) gives the aim of combining a small pressure-side blow holewith a short profile gap. As already mentioned however, the twocharacteristics behave in a contrary manner.

It is furthermore considered to be advantageous if main rotor (HR) andsecondary rotor (NR) are configured and tuned to one another in such amanner that a dry compression with a pressure ratio Π of up to 3, inparticular with a pressure ratio Π greater than 1 and up to 3 can beachieved, where the pressure ratio is the ratio of compression endpressure to suction pressure.

A further preferred embodiment provides a rotor pair in such a mannerthat the main rotor (HR) is configured to be operated relative to anaddendum circle KK₂ at a circumferential speed in a range from 20 to 100m/s.

A further embodiment provides a rotor pair which is characterized inthat for a diameter ratio defined by the ratio of the addendum circleradii of main rotor (HR) and secondary rotor (NR)

${{D_{v} = {\frac{{Dk}_{2}}{{Dk}_{1}} = \frac{{rk}_{2}}{{rk}_{1}}}}{{1.1}45}} \leq D_{v} \leq 1.30$

is maintained, where Dk₁ designates the diameter of the addendum circleKK₁ of the secondary rotor (NR) and Dk₂ designates the diameter of theaddendum circle KK₂ of the main rotor (HR).2. Preferred Embodiments for a Rotor Pair with Tooth-Number Ratio of 4/5

Preferred embodiments are presented hereinafter for a rotor pair havinga tooth number ratio of 4/5, i.e. for a rotor pair in which the mainrotor has four teeth and the secondary rotor has five teeth:

A further preferred embodiment provides that in a transverse sectionalview, circular arcs B₂₅, B₅₀, B₇₅ running within a secondary rotor toothare defined, the common centre point of which is given by the axis C1,wherein the radius r₂₅ of B₂₅ has the value r₂₅=rf₁+0.25*(rk₁−rf₁), theradius r₅₀ of B₅₀ has the value r₅₀=rf₁+0.5*(rk₁−rf₁), and the radiusr₇₅ of B₇₅ has the value r₇₅=rf₁+0.75*(rk₁−rf₁), and wherein thecircular arcs B₂₅, B₅₀, B₇₅ are each delimited by the leading toothflank F_(V) and trailing tooth flank F_(N), wherein tooth thicknessratios are defined as ratios of the arc lengths b₂₅, b₅₀, b₇₅ of thecircular arcs B₂₅, B₅₀, B₇₅ with ε₁=b₅₀/b₂₅ and ε₂=b₇₅/b₂₅ and thefollowing dimension is adhered to: 0.75≤ε₁≤0.85 and/or 0.65≤ε₂≤0.74.

The aim is to combine a small blow hole with short length of the profileengagement gap. However, the two parameters behave in a contrary manner,i.e. the smaller the blow hole is modelled, the larger the length of theprofile engagement gap must necessarily be. Conversely, the blow holebecomes larger, the shorter the length of the profile engagement gap. Inthe claimed ranges a particularly favourable combination of the twoparameters is achieved. At the same time, a sufficiently high flexuralrigidity of the secondary rotor is ensured. Furthermore, advantages areobtained as regards the chamber expulsion and the secondary rotortorque. With regard to the illustration of the parameters, reference isadditionally made to FIG. 7 c.

A further preferred embodiment provides that in a transverse sectionalview, foot points F1 and F2 are defined on the dedendum circle betweenthe observed tooth of the secondary rotor (NR) and the respectivelyadjacent tooth of the secondary rotor and an apex point F5 is defined atthe radially outermost point of the tooth, wherein a triangle D_(z) isdefined by F1, F2 and F5 and wherein in a radially outer region, thetooth projects beyond the triangle D_(z) with its leading tooth flankF_(V) formed between F5 and F2 with an area A1 and with its trailingtooth flank F_(N) formed between F1 and F5 with an area A2 and wherein6≤A2/A1≤15 is maintained.

The tooth sub-area A1 at the leading tooth flank F_(V) of the secondaryrotor has a substantial influence on the blow hole area. The toothsub-area A2 at the trailing tooth flank F_(N) of the secondary rotor onthe other hand has a substantial influence on the length of the profileengagement gap, the chamber expulsion and the secondary rotor torque.For the tooth sub-area ratio A2/A1 there is an advantageous range whichenables a good compromise between length of the profile engagement gapon the one hand and the blow hole on the other hand. With regard to theillustration of the parameters, reference is additionally made to FIG. 7d.

In a further embodiment, the rotor pair comprises a secondary rotor inwhich in a transverse sectional view, foot points F1 and F2 are definedbetween the observed tooth of the secondary rotor (NR) and therespectively adjacent tooth of the secondary rotor (NR), and an apexpoint F5 is defined at the radially outermost point of the tooth,wherein a triangle D_(z) is defined by F1, F2 and F5 and wherein in aradially outer region of the tooth, the leading tooth flank F_(V) formedbetween F5 and F2 projects with an area A1 beyond the triangle D_(z) andin a radially inner region is set back with respect to the triangleD_(z) with an area A3 and wherein 9.0≤A3/A1≤18 is maintained. Withregard to the illustration of the parameters, reference is additionallymade to FIG. 7 d.

Furthermore with regard to the configuration of the secondary rotor, itis considered to be advantageous if in a transverse sectional view, footpoints F1 and F2 are defined between the observed tooth of the secondaryrotor (NR) and the respectively adjacent tooth of the secondary rotor(NR) and an apex point F5 is defined at the radially outermost point ofthe tooth, wherein a triangle D_(z) is defined by F1, F2 and F5 andwherein in a radially outer region of the tooth, the leading tooth flankF_(V) formed between F5 and F2 projects with an area A1 beyond thetriangle D_(z), wherein the tooth itself has a cross-sectional area A0delimited by the circular arc B running between F1 and F2 about thecentre point defined by the axis C1 and wherein 1.5%≤A1/A0≤3.5% ismaintained.

With regard to the specification of the parameters, reference is made toFIGS. 7d and 7 e.

A further preferred embodiment provides that in a transverse sectionalview, foot points F1 and F2 are defined between the observed tooth ofthe secondary rotor (NR) and the respectively adjacent tooth of thesecondary rotor (NR) and an apex point F5 is defined at the radiallyoutermost point of the tooth, wherein the circular arc B running betweenF1 and F2 defines a tooth partition angle α corresponding to 360°/numberof teeth of the secondary rotor (NR) about the centre point defined bythe axis C1, wherein a point F11 is defined on the half circular arc Bbetween F1 and F2, wherein a radial half-line R drawn from the centrepoint of the secondary rotor (NR) defined by the axis C1 through theapex point F5 intersects the circular arc B at a point F12, wherein anoffset angle β is defined by the offset of F11 to F12 viewed in thedirection of rotation of the secondary rotor (NR) and wherein

14%≤δ≤18%

is maintained where

$\delta = {\frac{\beta}{\gamma}*{{100\;\lbrack\%\rbrack}.}}$

Firstly it is again clarified that the offset angle is preferably alwayspositive, i.e. the offset is always given in the direction of thedirection of rotation and not contrary to this. In this respect thetooth of the secondary rotor is curved with respect to the axis ofrotation of the secondary rotor. However, the offset should be kept in arange specified as advantageous in order to enable a favourablecompromise between the blow hole area, the shape of the engagement line,the length and the shape of the profile engagement gap, the secondaryrotor torque, the flexural rigidity of the rotors and the chamberexpulsion into the pressure window. With regard to the illustration ofthe parameters, reference is additionally made to FIG. 7 f.

It is furthermore considered to be advantageous if in a transversesectional view, the trailing tooth flank F_(N) of a tooth of thesecondary rotor (NR) formed between F1 and F5 has a convex lengthcomponent of at least 55% to at most 95%.

The relatively long convex length component of the trailing tooth flankF_(N) of a tooth of the secondary rotor specified with the range allowsa good compromise between length of the profile engagement gap, chamberexpulsion, secondary rotor torque on the one hand and flexural rigidityof the secondary rotor on the other hand. With regard to theillustration of the parameters, reference is additionally made to FIG. 7g.

Preferably the secondary rotor is configured such that in a transversesectional view, the radial half-line drawn from the axis C1 of thesecondary rotor (NR) through F5 divides the tooth profile into an areacomponent A5 assigned to the leading tooth flank F_(V) and an areacomponent A4 assigned to the trailing tooth flank F_(N) and wherein

4≤A4/A5≤9

is maintained. It should be noted once again at this point that thetooth profile is delimited radially inwards towards the C1 axis by thededendum circle FK₁. In this case, it can occur that the radialhalf-line R divides the tooth profile in such a manner that two disjointarea components with a total area component A5 which are assigned to theleading tooth flank F_(V) are formed, cf. FIG. 7g . If the apex point F5were to be offset with respect to the leading tooth flank in such amanner that the radial half-line F5 not only touches the leading toothflank F_(V) but intersects it at two points, two disjoint areacomponents assigned to the leading tooth flank with a total areacomponent A5 are again defined. The area component A4 assigned to thetrailing tooth flank F_(N) is then delimited on the one hand by theradial half-line R, in sections, namely between the two points ofintersection of the leading tooth flank F_(V) with the radial half-line,on the other hand by the leading tooth flank F_(V).

A further preferred embodiment comprises a rotor pair which ischaracterized in that the main rotor HR is formed with a wrap-aroundangle ϕ_(HR) for which it holds that: 320°≤ϕ_(HR)≤360°, preferably330°≤ϕ_(HR)≤360°.

With increasing wrap-around angle, the pressure window area can beconfigured to be larger for the same built-in volume ratio. In addition,the axial extension of the working chamber to be expelled, the so-calledprofile pocket depth, is shortened. This reduces the expulsion throttlelosses in particular at higher rotational speeds and thus enables abetter specific performance. A too-large wrap-around angle in turn has adisadvantageous effect on the installation volume and results in largerrotors.

In addition, in an advantageous embodiment a rotor pair is providedwhich is configured in such a manner and interacts with one another sothat a blow hole factor μ_(B1) is at least 0.02% and at most 0.4%,preferably at most 0.25%, wherein

$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\;\lbrack\%\rbrack}}$

and wherein A_(B1) designates an absolute pressure-side blow hole areaand A6 and A7 designate tooth gap areas of the secondary rotor (NR) orthe main rotor (HR), wherein the area A6 in a transverse sectional viewis the area enclosed between the profile course of the secondary rotor(NR) between two adjacent apex points F5 and the addendum circle KK₁ andthe area A7 in a transverse sectional view is the area enclosed betweenthe profile course of the main rotor (HR) between two adjacent apexpoints H5 and the addendum circle KK₂.

Whereas the absolute magnitude of the pressure-side blow hole alone doesnot allow any meaningful prediction about the effect on leakage massflows, a ratio of the absolute pressure-side blow hole area A_(B1) tothe sum of the tooth gap area A6 of the secondary rotor and the toothgap area A7 of the main rotor is substantially more predictive. Withregard to the further illustration of the parameters, reference isadditionally made here to FIG. 7b . The lower the numerical value ofμ_(B1), the smaller is the influence of the blow hole on the operatingbehaviour. The pressure-side blow hole area can thus be representedindependently of the installation size of the screw machine.

In a further preferred embodiment, a rotor pair is configured andmatched to one another in such a manner that

for a blow hole/profile gap length factor μ₁*μ_(B1) it holds that

0.1%≤μ₁*μ_(B1)≤1.72%

where

${\mu_{l} = \frac{l_{sp}}{{PT}_{1}}},$

where L_(sp) designates the length of the profile engagement gap of atooth gap of the secondary rotor and PT₁ designates the profile depth ofthe secondary rotor where PT₁=rk₁−rf₁and

$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\;\lbrack\%\rbrack}}$

where A_(B1) designates the absolute blow hole area and A6 and A7designate the profile areas of the secondary rotor (NR) or the mainrotor (HR), wherein the area A6 in a transverse sectional viewdesignates the area enclosed between the profile course of the secondaryrotor (NR) between two adjacent apex points F5 and the addendum circleKK₁, and the area A7 in a transverse sectional view designates the areaenclosed between the profile course of the main rotor (HR) between twoadjacent apex points H5 and the addendum circle KK₂.μ₁ designates a profile gap length factor, where a length of the profileengagement gap of a tooth gap is related to the profile depth PT₁. Thus,a measure for the length of the profile engagement gap can be specifiedindependently of the installation size of the screw machine, The lowerthe numerical value of the characteristic μ₁, the shorter is the profilegap for the same profile depth and therefore the smaller is the leakagevolume flow back to the suction side. The factor μ₁*μ_(B1) gives the aimof combining a small pressure-side blow hole with a short profile gap.As already mentioned however, the two characteristics behave in acontrary manner.

It is furthermore considered to be advantageous if main rotor (HR) andsecondary rotor (NR) are configured and tuned to one another in such amanner that a dry compression with a pressure ratio Π of up to 5, inparticular with a pressure ratio Π greater than 1 and up to 5 can beachieved, or alternatively a fluid-injected compression with a pressureratio Π of up to 16, in particular with a pressure ratio Π of greaterthan 1 and up to 16, where the pressure ratio is the ratio ofcompression end pressure to suction pressure.

A further preferred embodiment provides a rotor pair in such a mannerthat in the case of a dry compression the main rotor (HR) is configuredto be operated relative to an addendum circle KK₂ at a circumferentialspeed in a range from 20 to 100 m/s and in the case of a fluid-injectedcompression the main rotor (HR) is configured to be operated relative toan addendum circle KK₂ at a circumferential speed in a range from 5 to50 m/s.

A further embodiment comprises a rotor pair which is characterized inthat for a diameter ratio defined by the ratio of the addendum circleradii of main rotor (HR) and secondary rotor (NR)

$D_{v} = {\frac{{Dk}_{2}}{{Dk}_{1}} = \frac{{rk}_{2}}{{rk}_{1}}}$

it holds that where Dk₁ designates the diameter of the addendum circleKK₁ of the secondary rotor (NR) and Dk₂ designates the diameter of theaddendum circle KK₂ of the main rotor (HR).3. Preferred Embodiments for a Rotor Pair with a Tooth Number Ratio of5/6

Preferred embodiments are set out hereinafter for a rotor pair with atooth number ratio 5/6, i.e. for a rotor pair in which the main rotorhas five teeth and the secondary rotor has six teeth:

A first preferred embodiment provides that in a transverse sectionalview, circular arcs B₂₅, B₅₀, B₇₅ running within a secondary rotor toothare defined, the common centre point of which is given by the axis C1,wherein the radius r₂₅ of B₂₅ has the value r₂₅=rf₁+0.25*(rk₁−rf₁), theradius r₅₀ of B₅₀ has the value r₅₀=rf₁+0.5*(rk₁−rf₁), and the radiusr₇₅ of B₇₅ has the value r₇₅=rf₁+0.75*(rk₁−rf₁), and wherein thecircular arcs B₂₅, B₅₀, B₇₅ are each delimited by the leading toothflank F_(V) and trailing tooth flank F_(N), wherein tooth thicknessratios are defined as ratios of the arc lengths b₂₅, b₅₀, b₇₅ of thecircular arcs B₂₅, B₅₀, B₇₅ with ε₁=b₅₀/b₂₅ and ε₂=b₇₅/b₂₅ and thefollowing dimension is adhered to: 0.76≤ε₁<0.86 and/or 0.62≤ε₂≤0.72.

The aim is to combine a small blow hole with short length of the profileengagement gap. However the two parameters behave in a contrary manner,i.e. the smaller the blow hole is modelled, the larger the length of theprofile engagement gap necessarily becomes. Conversely the blow holebecomes larger, the shorter is the length of the profile engagement gap.In the claimed ranges a particularly favourable combination of the twoparameters is achieved. At the same time a sufficiently high flexuralrigidity of the secondary rotor is achieved. Furthermore, advantages areestablished as far as the chamber expulsion is concerned and for thesecondary rotor torque. With regard to the illustration of theparameters, reference is additionally made to FIG. 7 c.

A further preferred embodiment provides that in a transverse sectionalview, foot points F1 and F2 are defined on the dedendum circle betweenthe observed tooth of the secondary rotor (NR) and the respectivelyadjacent tooth of the secondary rotor and an apex point F5 is defined atthe radially outermost point of the tooth, wherein a triangle D_(z) isdefined by F1, F2 and F5 and wherein in a radially outer region, thetooth projects beyond the triangle D_(z) with its leading tooth flankF_(V) formed between F5 and F2 with an area A1 and with its trailingtooth flank F_(N) formed between F1 and F5 with an area A2 and wherein4≤A2/A1≤7 is maintained.

The tooth sub-area A1 at the leading tooth flank F_(V) of the secondaryrotor has a substantial influence on the blow hole area. The toothsub-area A2 at the trailing tooth flank F_(N) of the secondary rotor onthe other hand has a substantial influence on the length of the profileengagement gap, the chamber expulsion and the secondary rotor torque.For the tooth sub-area ratio A2/A1 there is an advantageous range whichenables a good compromise between length of the profile engagement gapon the one hand and the blow hole on the other hand. With regard to theillustration of the parameters, reference is additionally made to FIG. 7d.

In a further preferred embodiment, the rotor pair comprises a secondaryrotor in which in a transverse sectional view, foot points F1 and F2 aredefined between the observed tooth of the secondary rotor (NR) and therespectively adjacent tooth of the secondary rotor (NR) and an apexpoint F5 is defined at the radially outermost point of the tooth,wherein a triangle D_(z) is defined by F1, F2 and F5 and wherein in aradially outer region of the tooth, the leading tooth flank F_(V) formedbetween F5 and F2 projects with an area A1 beyond the triangle D_(z) andin a radially inner region is set back with respect to the triangleD_(z) with an area A3 and wherein 8.0≤A3/A1≤14 is maintained. Withregard to the illustration of the parameters, reference is additionallymade to FIG. 7 d.

Furthermore, with regard to the configuration of the rotor, it isconsidered to be advantageous if in a transverse sectional view, footpoints F1 and F2 are defined between the observed tooth of the secondaryrotor (NR) and the respectively adjacent tooth of the secondary rotor(NR) and an apex point F5 is defined at the radially outermost point ofthe tooth, wherein a triangle D_(z) is defined by F1, F2 and F5 andwherein in a radially outer region of the tooth, the leading tooth flankF_(V) formed between F5 and F2 projects with an area A1 beyond thetriangle D_(z), wherein the tooth itself has a cross-sectional area A0delimited by the circular arc B running between F1 and F2 about thecentre point defined by the axis C1 and wherein 1.9%≤A1/A0≤3.2% ismaintained. With regard to the illustration of the parameters, referenceis additionally made to FIGS. 7d and 7 e.

A further preferred embodiment provides that in a transverse sectionalview, foot points F1 and F2 are defined between the observed tooth ofthe secondary rotor (NR) and the respectively adjacent tooth of thesecondary rotor (NR) and an apex point F5 is defined at the radiallyoutermost point of the tooth, wherein the circular arc B running betweenF1 and F2 defines a tooth partition angle α corresponding to 360°/numberof teeth of the secondary rotor (NR) about the centre point defined bythe axis C1, wherein a point F11 is defined on the half circular arc Bbetween F1 and F2, wherein a radial half-line R drawn from the centrepoint of the secondary rotor (NR) defined by the axis C1 through theapex point F5 intersects the circular arc B at a point F12, wherein anoffset angle β is defined by the offset of F11 to F12 viewed in thedirection of rotation of the secondary rotor (NR) and wherein

13.5%≤δ≤18%

is maintained where

$\delta = {\frac{\beta}{\gamma}*{{100\;\lbrack\%\rbrack}.}}$

Firstly it is again clarified that the offset angle is preferably alwayspositive, i.e. the offset is always given in the direction of thedirection of rotation and not contrary to this. In this respect thetooth of the secondary rotor is curved with respect to the axis ofrotation of the secondary rotor. However, the offset should be kept in arange specified as advantageous in order to enable a favourablecompromise between the blow hole area, the shape of the engagement line,the length and the shape of the profile engagement gap, the secondaryrotor torque, the flexural rigidity of the rotors and the chamberexpulsion into the pressure window. With regard to the illustration ofthe parameters, reference is additionally made to FIG. 7 f.

A further preferred embodiment comprises a rotor pair which ischaracterized in that the main rotor HR is formed with a wrap-aroundangle ϕ_(HR) for which it holds that: 320°≤ϕ_(HR)≤360°, preferably330°≤ϕ_(HR)≤360°. With increasing wrap-around angle, the pressure windowarea can be configured to be larger for the same built-in volume ratio.In addition, the axial extension of the working chamber to be expelled,the so-called profile pocket depth, is shortened. This reduces theexpulsion throttle losses in particular at higher rotational speeds andthus enables a better specific performance. A too-large wrap-aroundangle in turn has a disadvantageous effect on the installation volumeand results in larger rotors.

In addition, in an advantageous embodiment a rotor pair is providedwhich is configured in such a manner and interacts with one another sothat a blow hole factor μ_(B1) is at least 0.03% and at most 0.25%,preferably at most 0.2%, wherein

$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\;\lbrack\%\rbrack}}$

and wherein A_(B1) designates an absolute pressure-side blow hole areaand A6 and A7 designate tooth gap areas of the secondary rotor (NR) orthe main rotor (HR), wherein the area A6 in a transverse sectional viewis the area enclosed between the profile course of the secondary rotor(NR) between two adjacent apex points F5 and the addendum circle KK₁ andthe area A7 in a transverse sectional view is the area enclosed betweenthe profile course of the main rotor (HR) between two adjacent apexpoints H5 and the addendum circle KK₂.

Whereas the absolute magnitude of the pressure-side blow hole alone doesnot allow any meaningful prediction about the effect on leakage massflows, a ratio of the absolute pressure-side blow hole area A_(B1) tothe sum of the tooth gap area A6 of the secondary rotor and the toothgap area A7 of the main rotor is substantially more predictive. Withregard to the further illustration of the parameters, reference isadditionally made here to FIG. 7b . The lower the numerical value ofμ_(B1), the smaller is the influence of the blow hole on the operatingbehaviour. The pressure-side blow hole area can thus be representedindependently of the installation size of the screw machine.

In a further preferred embodiment, a rotor pair is configured andmatched to one another in such a manner that for a blow hole/profile gaplength factor μ₁*μ_(B1) it holds that

0.1%≤μ₁*μ_(B1)≤1.26%

where

${\mu_{l} = \frac{l_{sp}}{{PT}_{1}}},$

where L_(sp) designates the length of the profile engagement gap of atooth gap of the secondary rotor and PT₁ designates the profile depth ofthe secondary rotor where PT₁=rk₁−rf₁and

$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\lbrack\%\rbrack}}$

where A_(B1) designates the absolute blow hole area and A6 and A7designate the profile areas of the secondary rotor (NR) or the mainrotor (HR), wherein the area A6 in a transverse sectional viewdesignates the area enclosed between the profile course of the secondaryrotor (NR) between two adjacent apex points F5 and the addendum circleKK₁, and the area A7 in a transverse sectional view designates the areaenclosed between the profile course of the main rotor (HR) between twoadjacent apex points H5 and the addendum circle KK₂.

μ₁ designates a profile gap length factor, where the length of theprofile engagement gap of a tooth gap is related to the profile depthPT₁. Thus, a measure for the length of the profile engagement gap can bespecified independently of the installation size of the screw machine.The lower the numerical value of the characteristic μ₁, the shorter isthe profile gap for the same profile depth and therefore the smaller isthe leakage volume flow back to the suction side. The factor μ₁*μ_(B1)gives the aim of combining a small pressure-side blow hole with a shortprofile gap. As already mentioned however, the two characteristicsbehave in a contrary manner.

It is furthermore considered to be advantageous if main rotor (HR) andsecondary rotor (NR) are configured and tuned to one another in such amanner that a dry compression with a pressure ratio Π of up to 5, inparticular with a pressure ratio Π greater than 1 and up to 5 can beachieved, or alternatively a fluid-injected compression with a pressureratio Π of up to 20, in particular with a pressure ratio Π of greaterthan 1 and up to 20, where the pressure ratio is the ratio ofcompression end pressure to suction pressure.

A further preferred embodiment provides a rotor pair in such a mannerthat in the case of a dry compression the main rotor (HR) is configuredto be operated relative to an addendum circle KK₂ at a circumferentialspeed in a range from 20 to 100 m/s and in the case of a fluid-injectedcompression the main rotor (HR) is configured to be operated relative toan addendum circle KK₂ at a circumferential speed in a range from 5 to50 m/s.

A further embodiment provides a rotor pair which is characterized inthat for a diameter ratio defined by the ratio of the addendum circleradii of main rotor (HR) and secondary rotor (NR) it holds that

${{D_{v} = {\frac{Dk_{2}}{Dk_{1}} = \frac{rk_{2}}{rk_{1}}}}{{1.1}9}} \leq D_{v} \leq 1.26$

where Dk₁ designates the diameter of the addendum circle KK₁ of thesecondary rotor (NR) and Dk₂ designates the diameter of the addendumcircle KK₂ of the main rotor (HR).

4. Preferred Embodiment for a Rotor Pair Having a Tooth-Number Ratio of3/4, 4/5 or 5/6

It is generally considered to be preferable that in a transversesectional view the teeth of the secondary rotor taper outwards, i.e. allcircular arcs running perpendicular to a radial half-line starting froma centre point defined by the axis C1, drawn through the point F5,decrease radially outwards starting from the trailing tooth flank F_(N)towards the leading tooth flank F_(V) in the sequence from F1 to F2 (orat least remain the same in sections). In other words, in a transversesectional view for all the arc lengths b(r), running inside a tooth ofthe secondary rotor, of the respectively appurtenant concentric circulararcs having the radius rf₁<r<rk₁ and the common central point defined bythe axis C1, which are each delimited by the leading tooth flank F_(V)and the trailing tooth flank F_(N), it holds that the arc lengths b(r)decrease monotonically with increasing radius r.

The teeth of the secondary rotor in this preferred embodiment aretherefore configured in such a manner that no constrictions areobtained, i.e. the width of one tooth of the secondary rotor does notincrease at any point but decreases radially outwards or remains at amaximum. This is considered to be appropriate in order to achieve on theone hand a small pressure-side blow hole with a nevertheless shortprofile engagement gap length.

Advantageously the transverse sectional configuration of the secondaryrotor (NR) is executed in such a manner that the direction of action ofthe torque which results from a reference pressure on the partialsurface of the secondary rotor delimiting the working chamber isdirected contrary to the direction of rotation of the secondary rotor.

Such a transverse sectional configuration has the effect that the entiretorque from the gas forces on the secondary rotor is directed contraryto the direction of rotation of the secondary rotor. As a result, adefined flank contact is achieved between the trailing secondary rotorflank F_(N) and the leading main rotor flank. This helps to avoid theproblem of so-called rotor rattling which can occur in unfavourable, inparticular non-steady-state operating situations. Rotor rattling isunderstood to be an advancement and lagging of the secondary rotorsuperimposed on the uniform rotational movement about its axis ofrotation which is accompanied by a rapidly changing impacting of thetrailing secondary rotor flanks against the leading main rotor flanksand then of the leading secondary rotor flanks against the trailing mainrotor flanks etc. This problem occurs in particular when the torque fromthe gas forces together with other torques (e.g. from bearing friction)on the secondary rotor is undefined (i.e. is close to zero, which iseffectively avoided by the advantageous transverse sectionalconfiguration.

In a specifically possible optional embodiment, main rotor (HR) andsecondary rotor (NR) are configured and tuned to one another forconveying air or inert gases such as helium or nitrogen.

It is preferred that in a transverse sectional view, the profile of atooth of the secondary rotor relative to the radial half-line R drawnfrom the centre point defined by the axis C1 through the apex point F5is configured to be asymmetrical. In the secondary rotor thereforeleading tooth flank and trailing tooth flank of each tooth areconfigured to be asymmetrical with respect to one another. Thisasymmetrical configuration is per se already known for screwcompressors. However, it makes a substantial contribution to efficientcompression.

A further preferred embodiment provides that in a transverse sectionalview a point C is defined on the connecting section C1C2 between thefirst axis (C1) and the second axis (C2) where the pitch circles WK₁ ofthe secondary rotor (NR) and WK₂ of the main rotor (HR) contact, that K5defines the point of intersection of the dedendum circle FK₁ of thesecondary rotor (NR) with the connecting section C1C2 , where r₁determines the distance between K5 and C and that K4 designates thepoint of the suction-side part of the line of engagement which lies atthe greatest distance from the connecting section C1C2 between C1 andC2, where r₂ determines the distance between K4 and C and where it holdthat:

${0.9} \leq \frac{r_{1}}{r_{2}} \leq {{{0.8}75 \times \frac{z_{1}}{z_{2}}} + {{0.2}2}}$

where z₁ is the number of teeth of the secondary rotor (NR) and z₂ isthe number of teeth of the main rotor (HR).

Inter alia, the secondary rotor torque (=torque on the secondary rotor)and the chamber expulsion into the pressure window can be influenced bymeans of the profile of the suction-side part of the line of engagementbetween the straight-line section C1C2 and the suction-side intersectionedge. Characteristic features of the aforesaid profile of thesuction-side part of the line of engagement can be described by means ofthe radii ratio r₁/r₂ of two concentric circles about the point C(=contact point of pitch circle WK₁ of the secondary rotor and pitchcircle WK₂ of the main rotor). If the radii ratio r₁/r₂ lies within thespecified range, the working chamber is expelled substantiallycompletely into the pressure window.

In a preferred embodiment, the rotor pair is formed and configured insuch a manner that for a rotor length ratio L_(HR)/a it holds that:0.85*(z₁/z₂)+0.67≤L_(HR)/a≤1.26*(z₁/z₂)+1.18, preferably0.89*(z₁/z₂)+0.94≤L_(HR)/a≤1.05*(z₁/z₂)+1.22, where z₁ is the number ofteeth of the secondary rotor (NR) and z₂ is the number of teeth of themain rotor (HR), wherein the rotor length ratio L_(HR)/a gives the ratioof the rotor length Lim to the axial distance a and rotor length L_(HR)is the distance of the suction-side main-rotor rotor end face to thepressure-side main-rotor rotor end face.

The lower the value of L_(HR)/a, the higher will be the flexuralrigidity of the rotors (for the same displacement). In the claimed rangethe flexural rigidity of the rotors is sufficiently high so that therotors do not bend significantly during operation and therefore the gap(between rotors or between rotors and compressor housing) can bedesigned to be relatively narrow without the risk thereby arising thatthe rotors run onto one another or run on in the compressor housingunder unfavourable operating conditions (high temperatures and/or highpressures). Narrow gaps offer the advantage of low back flows andtherefore contribute to the energy efficiency. At the same time, despitesmall gap dimensions, the operating safety is ensured. Also during rotormanufacture a high flexural rigidity of the rotors is advantageous foradhering to the high requirements for the shape tolerances.

On the other hand however, the ratio L_(HR)/a is so large that the axialdistance a is not excessively large in relation to the rotor lengthL_(HR). This is advantageous since in consequence the rotor diameter andquite specifically the end faces of the rotors are not excessivelylarge. As a result on the one hand, the gap lengths can be kept small;this results in a reduction of the back flow into preceding workingchambers and as a result in turn improvement of the energy efficiency.On the other hand, as a result of small end face dimensions, the axialforces resulting from the pressurized pressure-side end faces of therotors can advantageously be kept small, these axial forces act duringoperation on the rotors and in particular on the rotor mounting. Byminimizing these axial forces, the loading of the (roller) bearings canbe minimized or the bearings can have smaller dimensions.

It can advantageously be further provided that in a transverse sectionalview the tooth profile of the secondary rotor (NR) on its radially outersection in sections follows a circular arc ARC₁ having the radius rk₁,i.e. a plurality of points of the leading tooth flank F_(V) and thetrailing tooth flank F_(N) lie on the circular arc having the radius rk₁around the centre point defined by the axis C1, wherein preferably thecircular arc ARC₁ encloses an angle relative to the axis C1 between 0.5°and 5°, further preferably between 0.5° and 2.5°, wherein F10 is thepoint at the furthest distance from F5 on the leading tooth flank onthis circular arc and wherein the radial half-line R10 drawn between F10and the centre point of the secondary rotor (NR) defined by the axis C1contacts the leading tooth flank F_(V) at least at one point or at twopoints, cf. in particular the illustration in FIG. 7 h.

The previously described embodiment of the tooth profile of thesecondary rotor is primarily relevant for a tooth-number ratio of 3/4 or4/5. With such a tooth-number ratio, the blow hole area can be reducedby satisfying the condition reproduced above. For the tooth-number ratio5/6 on the other hand, an aforesaid contact point or aforesaid points ofintersection with the leading tooth flank F_(V), does not seem desirablesince the teeth of the secondary rotor then possibly become too thin andin consequence too flexible.

Furthermore a compressor block comprising a compressor housing and arotor pair as described previously is claimed according to theinvention, wherein the rotor pair comprises a main rotor HR and asecondary rotor NR, which are each mounted rotatably in the compressorhousing.

In a preferred embodiment, the compressor block is configured in such amanner that the transverse sectional configured is executed in such amanner that the working chamber formed between the tooth profiles ofmain rotor (HR) and secondary rotor (NR) can be expelled substantiallycompletely into the pressure window.

In general it is also considered to be advantageous that with theselection of the profiles of secondary rotor and main rotor presentedhere it is possible to completely dispense with a pressure-reliefgroove/noise groove or to make this small.

As a result of the transverse sectional configuration of the two rotors,it is advantageously achieved that during expulsion of the workingchambers into the pressure window, no chamber interstitial volume isformed between the two rotors. Compression can take place particularlyefficiently since no back flow of already-compressed medium to thesuction side takes place and with this no additional heat inputaccumulates. Furthermore, the entire compressed volume can be utilizedby downstream compressed air users. As a result, over-compression isavoided, advantages are obtained for the energy efficiency, for thesmooth running of the compressor block and for the lifetime of the rotorbearings. In oil-injected compressors, compression of the oil isprevented and thus the smooth running of the compressor is improved, theloading of the rotor mounting is reduced and the stressing of the oil isreduced.

In a further preferred embodiment a shaft end of the main rotor isguided out from the compressor housing and configured for connection toa drive, wherein preferably both shaft ends of the secondary rotor areaccommodated completely inside the compressor housing.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is explained in further detail hereinafter with regard tofurther features and advantages by reference to the description ofexemplary embodiments. In the figures:

FIG. 1 shows a transverse section of a first embodiment with atooth-number ratio of 3/4.

FIG. 2 shows a transverse section of a second embodiment with atooth-number ratio of 3/4.

FIG. 3 shows a transverse section of a third embodiment with atooth-number ratio of 4/5.

FIG. 4 shows a fourth exemplary embodiment in a transverse sectionalview with a tooth number ratio of 5/6.

FIG. 5 shows an illustration of the isentropic block efficiency for thesecond exemplary embodiment for the 3/4 tooth-number ratio compared withthe prior art.

FIG. 6 shows an illustration of the isentropic block efficiency for thefourth exemplary embodiment for the 5/6 tooth-number ratio compared withthe prior art.

FIG. 7a-7k shows illustration diagrams for the various parameters of thegeometry of the secondary rotor or the rotor pair consisting of mainrotor and secondary rotor.

FIG. 8 shows an illustration of the wrap-around angle at the main rotor.

FIG. 9 shows a schematic sectional drawing of an embodiment of acompressor block.

FIG. 10 shows an embodiment for an intermeshed rotor pair consisting ofa main rotor and a secondary rotor in three-dimensional view.

FIG. 11 shows a perspective view of one embodiment of a secondary rotorto illustrate the spatial line of engagement.

FIG. 12a, 12b shows an illustration of the areas or subareas of aworking chamber of one embodiment of the secondary rotor which arerelevant for the torque effects.

FIG. 13 shows the transverse section of the embodiment according to FIG.1 to explain the profile course of main and secondary rotor in thisembodiment.

FIG. 14 shows the transverse section of the embodiment according to FIG.2 to explain the profile course of main and secondary rotor in thisembodiment.

FIG. 15 shows the transverse section of the embodiment according to FIG.3 to explain the profile course of main and secondary rotor in thisembodiment.

FIG. 16 shows the transverse section of the embodiment according to FIG.4 to explain the profile course of main and secondary rotor in thisembodiment.

DETAILED DESCRIPTION

The exemplary embodiments according to FIGS. 1 to 4 will be explainedhereinafter.

All four exemplary embodiments represent suitable profiles in the senseof the present invention.

The corresponding geometrical specifications for the main rotor HR orthe secondary rotor NR are given in Tables 1 to 4 reproducedhereinafter.

TABLE 1 Exemplary Exemplary Exemplary Exemplary embodi- embodi- embodi-embodi- ment 1 ment 2 ment 3 ment 4 Teeth 3 3 4 5 number HR z₂ Teeth 4 45 6 number NR z₁ PT_(rel) [—] 0.588 0.54 0.528 0.455 a/rk₁ [—] 1.66 1.721.764 1.78

TABLE 2 The profiles were created with the following axial distances a:Exemplary Exemplary Exemplary Exemplary embodi- embod- embodi- embodi-ment 1 iment 2 ment 3 ment 4 Axial distance a 127 111 [mm]

TABLE 3 Thus the following transverse-section principal dimensions areobtained: Exemplary Exemplary Exemplary Exemplary embodi- embodi-embodi- embodi- ment ment 2 ment 3 ment 4 Dk₂ [mm] 191 186.1 186 154 Dk₁[mm] 153 147.7 144 124.7 rw₂ [mm] 54.4 56.4 50.5 rw₁ [mm] 72.6 70.6 60.5

TABLE 4 Further principal dimensions of the rotors: Exemplary ExemplaryExemplary Exemplary embodi- embodi- embodi- embodi- ment 1 ment 2 ment 3ment 4 Rotor length 307 293 235.5 L_(HR) [mm]

In the exemplary embodiments presented, the following features andcharacteristics according to the invention are obtained, which arepresented in Table 5:

TABLE 5 Compilation of the further features and characteristics:Exemplary Exemplary Exemplary Exemplary embodi- embodi- embodi- embodi-Feature ment 1 ment 2 ment 3 ment 4 Tooth thickness 0.85 0.82 0.80 0.79ratio ε₁ [—] Tooth thickness 0.74 0.64 0.69 0.65 ratio ε₂ [—] Area ratioA2/A1 15.7 37.8 10.0 6.2 [—] Area ratio A1/A0 2.3 1.1 2.2 2.3 [%] Arearatio A3/A1 9.9 19.6 12.6 11.6 [—] Tooth curvature 18.5 21.1 15.7% 15.2ratio δ [%] Convex length 66.9% 71.2% 62.7% — component [%] Radial toothThe tooth thickness of the secondary rotor thickness teeth decreasesmonotonically from the profile addendum circle radius rf₁ to thededendum circle radius rk₁ Radial half-line Radial half-line R₁₀ has twopoints of R₁₀ intersection with the leading tooth flank FV Area ratioA4/A5 7.5 10.1 5.5 — [—] Wrap-around 334.7° 330.3 330.3 angle Φ_(HR)μ_(B1) [%] 0.159 0.086 0.106 0.18 μ_(B1) * μ₁ [%] 0.94 0.53 0.631 1.058Profile transverse The working chamber can be expelled sectionalsubstantially completely into the pressure configuration in windowrelation to chamber expulsion Profile transverse The direction of actionof the NR torque sectional resulting from the gas forces is directedconfiguration in contrary to the direction of rotation of relation tothe secondary rotor secondary rotor torque Shape of 1.037 1.044 0.9841.0 engagement line r₁/r₂ Diameter ratio Dv 1.248 1.26 1.292 1.235 Rotorlength ratio 2.42 2.42 2.31 2.12 L_(HR)/a

The isentropic block efficiency compared to the prior art is illustratedfor the second exemplary embodiment for the 3/4 tooth-number ratio inFIG. 5. Two curves for the same pressure ratio are reproduced there. Thespecifically reproduced pressure ratio is 2.0 (ratio of output pressureto input pressure). The isentropic block efficiency could be improvedsignificantly compared with the values attainable with the prior art.

FIG. 6 shows the isentropic block efficiency compared to the prior artfor the fourth exemplary embodiment (5/6 tooth-number ratio). Two curvesfor the same pressure ratio are also reproduced here. The specificallyreproduced pressure ratio is 9.0 (ratio of output pressure to inputpressure). Here also the isentropic block efficiency could be improvedsignificantly compared with the values attainable with the prior art.

The quantity delivered specified in each case in FIGS. 5 and 6corresponds to the conveyed volume flow of the compressor block relativeto the suction state.

FIG. 7a shows in a transverse sectional view one embodiment forsecondary rotor NR and main rotor HR with the centre points given by thecorresponding axes C1 and C2. Furthermore, the geometrical principaldimensions or principal parameters of the transverse sectional view areshown:

-   -   Addendum circle KK₁ of the secondary rotor with appurtenant        addendum circle radius rk₁ or addendum circle diameter Dk₁    -   Addendum circle KK₂ of the main rotor with appurtenant addendum        circle radius rk₂ or addendum circle diameter Dk₂    -   Dedendum circle FK₁ of the secondary rotor with appurtenant        dedendum circle radius rf₁ or dedendum circle diameter Df₁    -   Dedendum circle FK₂ of the main rotor with appurtenant dedendum        circle radius rf₂ or dedendum circle diameter Df₂    -   Axial distance a between the first axis C1 and the second axis        C2    -   Pitch circle WK₁ of the secondary rotor with appurtenant pitch        circle radius rw₁ or pitch circle diameter D_(W1)    -   Pitch circle WK₂ of the main rotor with appurtenant pitch circle        radius rw₂ or pitch circle diameter D_(W2)

Also shown are the direction of rotation 24 of the secondary rotor andthe necessarily resulting direction of rotation of the main rotor duringoperation as a compressor.

The leading tooth flank F_(V) and the trailing tooth flank F_(N) arecharacterized on a secondary rotor tooth as representative for all teethof the secondary rotor. A tooth gap 23 is characterized asrepresentative of all tooth gaps of the secondary rotor. The profilecourse of the leading tooth flank F_(V) and of the trailing tooth flankF_(N) shown by reference to FIG. 7a corresponds to the exemplaryembodiment for a tooth-number ratio of 5/6 illustrated by reference toFIG. 4.

FIG. 7b shows in a transverse sectional view the tooth gap areas A6 andA7 as well as a side view of a blow hole. The profile courses shown inFIG. 7b to explain the tooth gap areas A6 and A7 correspond to theexemplary embodiment for a tooth number ratio of 3/4 illustrated byreference to FIG. 1.

Furthermore, FIG. 7b shows the position of the coordinate system of theblow hole area A_(B1) shown in FIG. 7k in relation to the rotor pair.

The coordinate system is spanned by the u-axis parallel to the rotor endfaces along the pressure-side intersection edge 11.

The pressure-side blow hole lies in the described coordinate system andquite specifically in a plane perpendicular to the rotor end facesbetween the pressure-side intersection edge 11 and an engagement linepoint K2 of the pressure-side part of the line of engagement.

In a transverse sectional view the line of engagement 10 is divided intotwo sections by the connecting line between the two centre points C1 andC2: the suction-side part of the line of engagement is shown below, thepressure-side part is shown above the connecting line.

K2 designates the point of the pressure-side part of the line ofengagement 10 which lies at the furthest distance from the straightlines through C1 and C2. As a result of the intersection of the addendumcircles of the two rotors, a pressure-side intersection edge 11 and asuction-side intersection edge 12 are formed. In FIG. 7b thepressure-side intersection edge 11 is shown as a point in a transversesectional view. The same applies to the depiction of the suction-sideintersection edge 12.

The u-axis is a parallel to the rotor end faces and in a transversesectional view corresponds to the vector from the engagement line pointK2 to the pressure-side intersection edge 11. Further details on thepressure-side blow hole area A_(B1) are obtained from FIG. 7 k.

FIG. 7c shows in a transverse sectional view a tooth of the secondaryrotor with the concentric circular arcs B₂₅, B₅₀, B₇₅ running inside therotor tooth around the centre point C1 with the appurtenant radii R₂₅,B₅₀, r₇₅ and the appurtenant arc lengths b₂₅, b₅₀, b₇₅.

The circular arcs B₂₅, B₅₀, B₇₅ are in each case delimited by theleading tooth flank F_(V) and the trailing tooth flank F_(N). Theprofile course of the leading tooth flank F_(V) and the trailing toothflank F_(N) shown by reference to FIG. 7c corresponds to the exemplaryembodiment explained by reference to FIG. 4 for a tooth-number ratio of5/6.

FIG. 7d shows in a transverse sectional view foot points F1 and F2 onthe addendum circle between the observed tooth of the secondary rotorand the respectively adjacent tooth of the secondary rotor and an apexpoint F5 at the radially outermost point of the tooth. Furthermore, thetriangle D_(z) defined by the points F1, F2 and F5 is shown.

FIG. 7d shows the following (tooth sub-)areas:

Tooth sub-area A1 corresponds to the area with which the observed toothprojects with its leading tooth flank F_(V) formed between F5 and F2beyond the triangle D_(z) in a radially outer region.

Tooth sub-area A2 corresponds to the area with which the observed toothprojects with its trailing tooth flank F_(N) formed between F5 and F1beyond the triangle D_(z) in a radially outer region.

Area A3 corresponds to the area with which the observed tooth is setback with its leading tooth flank formed between F5 and F2 with respectto the triangle D_(z).

Also shown is the tooth partition angle γ corresponding to 360°/numberof teeth of the secondary rotor. The profile course of the leading toothflank F_(V) and the trailing tooth flank F_(N) shown by reference toFIG. 7d corresponds to the exemplary embodiment explained by referenceto FIG. 4 for a tooth-number ratio of 5/6.

FIG. 7e shows in a transverse sectional view the cross-sectional area A0of a tooth of the secondary rotor which is delimited by the circular arcB running between F1 and F2 about the centre point C1. The profilecourse of the leading tooth flank F_(V) and the trailing tooth flankF_(N) shown by reference to FIG. 7e corresponds to the exemplaryembodiment explained by reference to FIG. 4 for a tooth-number ratio of5/6.

FIG. 7f shows in a transverse sectional view the offset angle (3. Thisis defined by the offset from point F1 l to point F12 observed in thedirection of rotation of the secondary rotor. Flt is a point on the halfcircular arc B between F1 and F2 about the centre point C1 andconsequently corresponds to the point of intersection of the anglebisector of the tooth partition angle γ with the circular arc B.

F12 is obtained from the point of intersection of the radial half-line Rdrawn from the centre point C1 to the apex point F5 with the circulararc B. The profile course of the leading tooth flank F_(V) and thetrailing tooth flank F_(N) shown by reference to FIG. 7f corresponds tothe exemplary embodiment explained by reference to FIG. 4 for atooth-number ratio of 5/6.

FIG. 7g shows in a transverse sectional view the turning point F8 on thetrailing tooth flank F_(N) of the secondary rotor at which the curvatureof the course of the tooth profile changes between addendum and dedendumcircle.

The trailing tooth flank F_(N) of the secondary rotor is divided by thepoint F8 into a substantially convexly curved component between F8 andthe apex point F5 and a substantially concavely curved component betweenF8 and the foot point F1.

FIG. 7h shows in a transverse sectional view two points of intersectionof the radial half-line R₁₀ from C1 to F10 with the leading tooth flankF_(V) of the secondary rotor, wherein the point F10 designates thatpoint of the leading tooth flank F_(V) which lies on the addendum circleKK₁ and is at the furthest distance from F5. The tooth flank thereforeradially outwards over a defined section follows a circular arc ARC1with radius rk₁ about the centre point of the secondary rotor defined bythe axis C1. The profile courses of the leading tooth flank F_(V) andthe trailing tooth flank F_(N) explained by reference to FIG. 7hcorrespond to the exemplary embodiment according to FIG. 1 for atooth-number ratio of 3/4.

FIG. 7i shows in a transverse sectional view the tooth profile dividedby the radial half-line drawn from C1 to F5.

Specifically in the embodiment shown, the tooth profile is divided intoan area component A4 assigned to the trailing tooth flank F_(N) and anarea component A5 assigned to the leading tooth flank F_(V). The profilecourses of the leading tooth flank F_(V) and the trailing tooth flankF_(N) explained by reference to FIG. 7i correspond to the exemplaryembodiment according to FIG. 4 described for a tooth-number ratio of5/6.

FIG. 7j shows in a transverse sectional view the line of engagement 10between main and secondary rotor as well as the two concentric circlesabout the point C having the radii r₁ and r₂ to describe thecharacteristic features of the course of the suction-side part of theline of engagement.

The line of engagement 10 is divided into two sections by the connectingsection between the first axis C1 and the second axis C2: thesuction-side part of the line of engagement is shown below, thepressure-side part is shown above the connecting section C1C2 .

Point C is the point of contact of the pitch circle WK₁ of the secondaryrotor with the pitch circle WK₂ of the main rotor.

K4 designates the point of the suction-side part of the line ofengagement which lies at the greatest distance from the connectingsection between C1 and C2.

Radius r₁ is the distance between K5 and C, radius r₂ designates thedistance between K4 and C.

FIG. 7 k:

FIG. 7k shows a pressure-side blow hole area A_(B1) of a working chamberand specifically in a sectional view perpendicular to the rotor endfaces. The delimitation of the blow hole area A_(B1) is formed here fromthe line of intersection 27 of the above-described imaginary flatsurface with the leading secondary-rotor tooth flank F_(v), the line ofintersection 26 of the plane with the trailing HR flank and a straightline section [K1 K3] of the pressure-side intersection edge 11.

The coordinate system of the pressure-side blow hole lies in the flatsurface described in FIG. 7b and is spanned by

-   -   the u-axis parallel to the rotor end faces (vector from the        engagement line point K2 to the pressure-side intersection edge        11) and    -   the pressure-side intersection edge 11.

In FIG. 8 the wrap-around angle Φ already discussed several times isillustrated once again. Specifically this is the angle Φ through whichthe transverse section is turned from the suction-side to thepressure-side rotor end face. This is illustrated in the present case bythe turning of the profile between a pressure-side end face 13 and asuction-side end face 14 through the angle Φ_(HR) at the main rotor HR.

FIG. 9 shows a schematic sectional view of a compressor block 19comprising a housing 15 as well as two rotors toothed with one anotherin pairs, mounted therein, namely a main rotor HR and a secondary rotorNR. Main rotor HR and secondary rotor NR are each mounted rotatably in ahousing 15 by means of suitable bearings 16. A drive power can beapplied to a shaft 17 of the main rotor HR, for example with a motor(not shown) via a coupling 18.

The compressor block shown is an oil-injected screw compressor in whichthe torque transmission between main rotor HR and secondary rotor NR isaccomplished directly by means of the rotor flanks. In contrast to thisin a dry screw compressor any contact of the rotor flanks can be avoidedby means of a synchronization transmission (not shown).

Also not shown are a suction connection for suction of the medium to becompressed and an outlet for the compressed medium.

FIG. 10 shows intermeshed main rotor HR and secondary rotor NR in aperspective view.

FIG. 11 shows the spatial line of engagement 10 of precisely one toothgap 23. The profile gap length I_(sp) is the length of the spatial lineof engagement of precisely one tooth gap 23. This therefore correspondsto the profile gap length of precisely one tooth pitch.

The entire torque of the gas forces on the secondary rotor is composedof the sum of the torque effects of the gas pressures in all workingchambers on the sub-surfaces of the secondary rotor delimiting therespective working chambers. In FIG. 12a such a sub-surface (22) of thesecondary rotor delimiting a working chamber is shown hatched as anexample.

FIG. 12b shows the division of the sub-surface (22) delimiting a workingchamber, shown in FIG. 12a into an area (28) shown dotted and an area(29) shown cross-hatched. Only the cross-hatched area (29) makes acontribution to the torque.

The sub-surface (22) is obtained from the specific transverse sectionalconfiguration and pitch of the secondary rotor. The pitch of thesecondary rotor relates to the pitch of the screw-shaped toothedstructure of the secondary rotor. The three-dimensional line ofengagement (10) delimiting the sub-surface, also shown in FIG. 12a isalso specified by the transverse sectional configuration of thesecondary rotor and the pitch.

Sub-surface (22) is also delimited by line of intersection (27). Detailson the line of intersection (27) have already been presented anddescribed within the framework of FIGS. 7b and 7k . The same applies tothe engagement line point K2.

The specific length of a working chamber in the direction of the axis ofrotation, which is dependent on the angular position of the secondaryrotor with respect to the main rotor, between the secondary rotor endface (20) on the one hand and the delimitation by the three-dimensionalline of engagement (10) and line of intersection (27) on the other handdoes not play any significant role here because—as is described in therelevant literature—the gas pressures on regions of the rotor surfacewhich in a sectional plane perpendicular to the axis of the rotorcorrespond to complete tooth gaps (shown dotted in FIG. 12b ) make nocontribution to the torque. The pitch of the secondary rotor only has aneffect on the magnitude but not on the direction of action of thetorque.

The area (28) shown dotted in FIG. 12b and the area (29) showncross-hatched in FIG. 12b together form the sub-surface (22).

Only the area (29) shown cross-hatched in FIG. 12b makes a contributionto the torque.

Thus, in each working chamber, the direction of action of the torquewhich is brought about by the gas pressure in the working chamber (or anarbitrary reference pressure) on the sub-surface of the secondary rotordelimiting the working chamber, is specified by the transverse sectionalconfiguration of the secondary rotor.

The above-described advantageous transverse sectional configuration ofthe secondary rotor (NR) thus results for each sub-surface (22) of thesecondary rotor delimiting a working chamber and thus for the entiresecondary rotor in a direction of action (25) of the torque from the gasforces which is directed contrary to the direction of rotation (24) ofthe secondary rotor, whereby rotor rattling is effectively avoided.

The exemplary embodiments presented confirm that with the presentinvention a considerable increase in efficiency could be achieved for arotor pair used in screw machines consisting of main rotor and secondaryrotor having a corresponding profile geometry.

With the present invention it has been possible to further improve theefficiency and smooth running of rotor profiles compared with the priorart independently of a specifically claimed profile definition.

Although it will easily be possible for the person skilled in the artusing the specified parameter values to produce suitable profile coursesusing conventional methods in the prior art, purely as an example theprofile courses in the previously discussed exemplary embodimentsaccording to FIGS. 1 to 4 will be explained in detail hereinafter. As isbest known to the person skilled in the art working in the presentfield, in order to generate profile courses, profile courses can also begenerated using publicly accessible computer programs.

Purely as an example in this connection mention is made of SV_Win, aproject of Vienna Technical University, where this software is describedin great detail in the Grafinger post-doctoral thesis. An alternative,publicly accessible computer program is moreover the DISCO software andin particular the SCORPATH module of the City University London (Centrefor Positive Displacement Compressor Technology). General information onthis can be obtained from: http://www.city.compressors.co.uk/.Information on installation of the software can be obtained fromhttp://www.staff.city.ac.uk/˜ra600?DISCO/DISCO/Installation%20instructions.pdf.A preview of the DISCO software can be found athttp://www.staff.city.ac.uk/˜ra600/DISC/DISCO%20Preview.htm.

Another alternative software is the software ScrewView which is alsomentioned in the thesis “Directed Evolutionary Algorithms” by StefanBerlik, Dortmund 2006 (p. 173 f). On the internet pagehttp://pi.information.uni-siegen.de/Mitarbeiter/berlip/projekte/ theScrewView software is described in detail in connection with the project“Method for the design of dry-running rotary compressor machines.”

In FIGS. 13 to 16 a tooth with trailing rotor flank F_(N) and leadingrotor flank F_(V) is specifically produced as follows: the section S1 toS2 is obtained from a circular arc on the secondary rotor NR about thecentre point C1 produced by the circular arc section T1 to T2 about thecentre point C2 on the main rotor HR. The section S2 to S3 is obtainedfrom an envelope curve to a trochoid produced by circular arc section T2to T3 about the centre point M4 on the main rotor HR. The section S3 toS4 is defined by a circular arc about the centre point M1. The sectionS4 to S5 is predefined by a circular arc about the centre point M2.

The section S5 to S6 is specified by a circular arc about the centrepoint C1. The adjoining section S6 to S7 is predefined by a circular arcabout the centre point M3. The section S7 to S1 is finally predefined byan envelope curve to a trochoid produced by the circular arc section T7to T1 about the centre point M5 on the main rotor HR. The previouslydescribed sections each adjoin one another seamlessly in the specifiedsequence. The tangents at the end of one section and at the beginning ofthe adjacent section are each the same. The sections in this respectmerge into one another directly, smoothly and free from bends.

The profile course of the teeth of the main rotor HR is explainedbriefly hereinafter for the exemplary embodiment according to FIGS. 1 to4 also with reference to FIGS. 13 to 16. The section T1-T2 is obtainedby a circular arc on the main rotor HR about the centre point C2 on themain rotor HR. The section T2-T3 is defined by the circular arc on themain rotor HR about the centre point M4. The section T3-T4 is obtainedfrom an envelope curve to a trochoid produced by the section S3-S4 onthe secondary rotor NR. The section T4-T5 is predefined by an envelopecurve to a trochoid produced by the section S4-S5 on the secondaryrotor. The section T5-T6 is defined by a circular arc about the centrepoint C2 produced by the circular arc section S5-S6 about the centrepoint C1 on the secondary rotor NR. The section T6-T7 is obtained by anenvelope curve to a trochoid produced by the section S6-S7 on thesecondary rotor NR. The section T7-T1 finally is specified by a circulararc about the centre point M5. Here it also applies that: the previouslydescribed sections each adjoin one another seamlessly in the specifiedsequence. The tangents at the end of one section and at the beginning ofthe adjacent section are each the same. The sections in this respectmerge into one another directly, smoothly and free from bends.

In general it should be noted that the profile courses of secondaryrotor NR and main rotor HR are naturally matched to one another and inthis respect the envelope curves to a trochoid each correspond tocircular arc sections on the counter-rotor. Furthermore, as alreadymentioned a tangential transition from one to the next section isensured. A general procedure for calculating the profile course of thecounter rotor is described for example in the Helpertz thesis “Methodfor stochastic optimization of screw rotor profiles”, Dortmund 2003, p.60 ff.

1. A rotor pair for a compressor block of a screw machine, wherein therotor pair comprises: a secondary rotor that rotates about a first axisand a main rotor that rotates about a second axis, wherein a number ofteeth of the main rotor is 3 and a number of teeth of the secondaryrotor is 4, wherein a relative profile depth of the secondary rotor${PT_{rel}} = \frac{{rk_{1}} - {rf_{1}}}{rk_{1}}$ is between 0.50 and0.65, wherein rk₁ is an addendum circle radius drawn around the outercircumference of the secondary rotor and rf₁ is a dedendum circle radiusstarting at the profile base of the secondary rotor, wherein a ratio ofan axis distance a of the first axis from the second axis and theaddendum circle radius rk₁ $\frac{a}{{rk}_{1}}$ is between 1.636 and1.8, wherein optionally the main rotor is configured with a wrap-aroundangle ϕ_(HR) for which it holds that 240°≤ϕ_(HR)≤360°, and whereinpreferably for a rotor length ratio L_(HR)/a:1.4≤L _(HR) /a≤3.4, wherein optionally the rotor length ratio is formedfrom a ratio of the rotor length L_(HR) of the main rotor and an axisdistance a, and the rotor length L_(HR) of the main rotor is formed by adistance of a suction-side main-rotor rotor end face to an oppositepressure-side main-rotor rotor end face. 2-14. (canceled)
 15. A rotorpair for a compressor block of a screw machine, wherein the rotor paircomprises a secondary rotor that rotates about a first axis and a mainrotor that rotates about a second axis, wherein a number of teeth of themain rotor is 4 and the number of teeth of the secondary rotor is 5,wherein a relative profile depth of the secondary rotor${PT_{rel}} = \frac{{rk_{1}} - {rf_{1}}}{rk_{1}}$ is at least 0.5,optionally at least 0.515, and at most 0.58 wherein rk₁ is an addendumcircle radius drawn around an outer circumference of the secondary rotorand rf₁ is a dedendum circle radius starting at a profile base of thesecondary rotor, wherein a ratio of an axis distance a of a first axisfrom a second axis and the addendum circle radius rk₁$\frac{a}{{rk}_{1}}$ is between 1.683 to 1.836, wherein optionally themain rotor is configured with a wrap-around angle ϕ_(HR) for which itholds that 240°≤ϕ_(HR)≤360°, and wherein optionally for a rotor lengthratio L_(HR)/a:1.4≤L _(HR) /a≤3.3, wherein a rotor length ratio is formed from a ratioof the rotor length L_(HR) of the main rotor and the axis distance a andthe rotor length L_(HR) of the main rotor is formed by a distance of asuction-side main-rotor rotor end face to an opposite pressure-sidemain-rotor rotor end face. 16-28. (canceled)
 29. A rotor pair for acompressor block of a screw machine, comprising a secondary rotor thatrotates about a first axis and a main rotor that rotates about a secondaxis, wherein a number of teeth of the main rotor is 5 and a number ofteeth of the secondary rotor is 6, wherein a relative profile depth ofthe secondary rotor ${PT_{rel}} = \frac{{rk_{1}} - {rf_{1}}}{rk_{1}}$ isbetween 0.44 and 0.495 wherein rk₁ is an addendum circle radius drawnaround an outer circumference of the secondary rotor and rf₁ is adedendum circle radius starting at a profile base of the secondaryrotor, wherein a ratio of the axis distance a of the first axis from thesecond axis and an addendum circle radius rk₁ $\frac{a}{{rk}_{1}}$ isbetween 1.74 and 1.8, wherein optionally the main rotor is configuredwith a wrap-around angle ϕ_(HR) for which 240°≤ϕ_(HR)≤360°, and whereinoptionally for a rotor length ratio L_(HR)/a1.4≤L _(HR) /a≤3.2, wherein a rotor length ratio is formed from a ratioof a rotor length Lim of the main rotor and the axis distance a and arotor length Lim of the main rotor is formed by a distance of asuction-side main-rotor rotor end face to an opposite pressure-sidemain-rotor rotor end face.
 30. The rotor pair according to claim 29,wherein in a transverse sectional view, circular arcs B₂₅, B₅₀, B₇₅running within a secondary rotor tooth are defined, the common centrepoint of which is given by the first axis, wherein a radius r₂₅ of B₂₅has a value r₂₅=rf₁+0.25*(rk₁−rf₁), a radius r₅₀ of B₅₀ has a valuer₅₀=rf₁+0.5*(rk₁−rf₁), and a radius r₇₅ of B₇₅ has a valuer₇₅=rf₁+0.75*(rk₁−rf₁), and wherein the circular arcs B₂₅, B₅₀, B₇₅ areeach delimited by a leading tooth flank F_(V) and a trailing tooth flankF_(N), wherein tooth thickness ratios are defined as ratios of the arclengths b₂₅, b₅₀, b₇₅ of the circular arcs B₂₅, B₅₀, B₇₅ with ε₁=b₅₀/b₂₅and ε₂=b₇₅/b₂₅ and at least one of 0.76<ε₁≤0.86 and/or 0.62≤ε₂≤0.72. 31.The rotor pair according to claim 29, wherein in a transverse sectionalview, foot points F1 and F2 are defined between an observed tooth of asecondary rotor and the respectively adjacent tooth of the secondaryrotor and an apex point F5 is defined at a radially outermost point ofthe observed tooth, wherein a triangle D_(z) is defined by foot pointsF1, F2 and apex point F5 and wherein in a radially outer region, theobserved tooth projects beyond the triangle D_(z) with its leading toothflank F_(V) formed between apex point F5 and F2 with an area A1 and withits trailing tooth flank F_(N) formed between foot point F1 and apexpoint F5 with an area A2 and wherein 4≤A2/A1≤7.
 32. The rotor pairaccording to claim 29, wherein in a transverse sectional view, footpoints F1 and F2 are defined between an observed tooth of the secondaryrotor and a respectively adjacent tooth of the secondary rotor and anapex point F5 is defined at a radially outermost point of the observedtooth, wherein a triangle D_(z) is defined by foot points F1, F2 andapex point F5 and wherein in a radially outer region of the observedtooth, a leading tooth flank F_(V) formed between apex point F5 and footpoint F2 projects with an area A1 beyond the triangle D_(z) and in aradially inner region is set back with respect to the triangle D_(z)with an area A3 and wherein 8≤A3/A1≤14.
 33. The rotor pair according toclaim 29, wherein in a transverse sectional view, foot points F1 and F2are defined between an observed tooth of the secondary rotor and arespectively adjacent tooth of the secondary rotor and an apex point F5is defined at a radially outermost point of the observed tooth, whereina triangle D_(z) is defined by foot points F1, F2 and apex point F5 andwherein in a radially outer region of the observed tooth, a leadingtooth flank F_(V) formed between apex point F5 and foot point F2projects with an area A1 beyond the triangle D_(z), wherein the toothitself has a cross-sectional area A0 delimited by a circular arc Brunning between foot points F1 and F2 about a centre point defined bythe first axis and wherein 1.9%≤A1/A0≤3.2%.
 34. The rotor pair accordingto claim 29, wherein in a transverse sectional view, foot points F1 andF2 are defined between an observed tooth of the secondary rotor and arespectively adjacent tooth of a secondary rotor and an apex point F5 isdefined at a radially outermost point of the observed tooth, wherein thecircular arc B running between foot points F1 and F2 defines a toothpartition angle γ corresponding to 360°/number of teeth of the secondaryrotor about a centre point defined by the first axis, wherein a point F1t is defined on the half circular arc B between foot points F1 and F2,wherein a radial half-line R drawn from a centre point of the secondaryrotor defined by the first axis through the apex point F5 intersects acircular arc B at a point F12, wherein an offset angle (3 is defined byan offset of points F11 to F12 viewed in a direction of rotation of thesecondary rotor and wherein13.5%≤δ≤18% where$\delta = {\frac{\beta}{\gamma}*{{100\lbrack\%\rbrack}.}}$
 35. The rotorpair according to claim 29, wherein the main rotor is formed with awrap-around angle ϕ_(HR) for which 320°≤ϕ_(HR)≤360°.
 36. The rotor pairaccording claim 29, wherein a blow hole factor μ_(B1) is between 0.03%and 0.25%, wherein$\mu_{Bl} = {\frac{A_{Bl}}{{A6} + {A7}}*{100\lbrack\%\rbrack}}$ andwherein A_(B1) designates an absolute pressure-side blow hole area andA6 and A7 designate tooth gap areas of the secondary rotor or the mainrotor, wherein an area A6 in a transverse sectional view is an areaenclosed between a profile course of the secondary rotor between twoadjacent apex points F5 and an addendum circle KK₁ and an area A7 in atransverse sectional view is an area enclosed between a profile courseof the main rotor between two adjacent apex points H5 and an addendumcircle KK₂.
 37. The rotor pair according to claim 29, wherein for a blowhole/profile gap length factor μ₁*μ_(B1)0.1%≤μ₁*μ_(B1)≤1.26% where ${\mu_{l} = \frac{l_{sp}}{{PT}_{1}}},$ wherel_(sp) designates a length of a profile engagement gap of a tooth gap ofthe secondary rotor and PT₁ designates a profile depth of the secondaryrotor where PT₁=rk₁ rf₁ and$\mu_{Bl} = {\frac{A_{Bl}}{{A\; 6} + {A\; 7}}*{100\lbrack\%\rbrack}}$where A_(B1) designates an absolute blow hole area and A6 and A7designate a profile areas of the secondary rotor or the main rotor,wherein an area A6 in a transverse sectional view designates a areaenclosed between a profile course of the secondary rotor between twoadjacent apex points F5 and an addendum circle KK₁, and an area A7 in atransverse sectional view designates an area enclosed between a profilecourse of the main rotor between two adjacent apex points H5 and anaddendum circle KK₂.
 38. The rotor pair according to claim 29, whereinthe main rotor and the secondary rotor are configured and tuned to oneanother in such a manner that a dry compression with a pressure ratio Πof up to 5 is achieved, or alternatively a fluid-injected compressionwith a pressure ratio Π of up to 20 where a pressure ratio is a ratio ofcompression end pressure to suction pressure.
 39. The rotor pairaccording to claim 29, wherein dry compression the main rotor isconfigured to be operated relative to an addendum circle KK₂ at acircumferential speed in a range from 20 to 100 m/s and a fluid-injectedcompression the main rotor is configured to be operated relative to anaddendum circle KK₂ at a circumferential speed in a range from 5 to 50m/s.
 40. The rotor pair according claim 29, wherein for a diameter ratiodefined by a ratio of an addendum circle radii of the main rotor and thesecondary rotor$D_{v} = {\frac{Dk_{2}}{Dk_{1}} = \frac{rk_{2}}{rk_{1}}}$1.19 ≤ D_(v) ≤ 1.26 where Dk₁ designates a diameter of an addendumcircle KK₁ of the secondary rotor and Dk₂ designates a diameter of anaddendum circle KK₂ of the main rotor.
 42. The rotor pair according toclaim 29, wherein a transverse sectional view arc lengths b(r), runninginside a tooth of the secondary rotor, of a respectively appurtenantconcentric circular arcs having a radius rf₁<r<rk₁ and a common centralpoint defined by the first axis are each delimited by a leading toothflank F_(V) and a trailing tooth flank F_(N) and the arc lengths b(r)decrease monotonically with increasing radius r.
 43. The rotor pairaccording to claim 29, wherein a transverse sectional configuration ofthe secondary rotor is executed in such a manner that a direction ofaction of torque which results from a reference pressure on a partialsurface of the secondary rotor delimiting a working chamber is directedcontrary to the direction of rotation of the secondary rotor.
 44. Therotor pair according to claim 29, wherein the main rotor and secondaryrotor are configured and tuned to one another for conveying air or inertgases.
 45. The rotor pair according to claim 29, wherein in a transversesectional view, the profile of a tooth of the secondary rotor relativeto the radial half-line R drawn from the centre point defined by thefirst axis C1 through the apex point F5 is configured to beasymmetrical.
 46. The rotor pair according to claim 29, wherein in atransverse sectional view a point C is defined on a connecting sectionbetween the first axis and the second axis where a pitch circles WK₁ ofthe secondary rotor and WK₂ of the main rotor contact, that K5 defines apoint of intersection of a dedendum circle FK₁ of the secondary rotorwith the connecting section where r₁ determines the distance between K5and C and that K4 designates a point of the suction-side part of a lineof engagement which lies at a greatest distance from the connectingsection between the first and second axis, where r₂ determines adistance between K4 and C and where:${0.9} \leq \frac{r_{1}}{r_{2}} \leq {{{0.8}75 \times \frac{z_{1}}{z_{2}}} + {{0.2}2}}$where z₁ is a number of teeth of the secondary rotor and z₂ is a numberof teeth of the main rotor.
 47. The rotor pair according claim 29,wherein for a rotor length ratio L_(HR)/a it holds:0.85*(z₁/z₂)+0.67≤L_(HR)/a≤1.26*(z₁/z₂)+1.18 where z₁ is a number ofteeth of the secondary rotor and z₂ is a number of teeth of the mainrotor, wherein a rotor length ratio L_(HR)/a denotes a ratio of a rotorlength L_(HR) to the axial distance a and the rotor length L_(HR) is thedistance of a suction-side main-rotor rotor end face to thepressure-side main-rotor rotor end face.
 48. A compressor blockcomprising a compressor housing and a rotor pair according to claim 29,wherein the rotor pair comprises the main rotor and the secondary rotor,which are each mounted rotatably in the compressor housing.